V. I. Arnold Dynamics, Statistics and Projective Geometry of Galois Fields

Dynamics, Statistics and Projective Geometry of Galois FieldsV.I. Arnold

2 Abstract. This book represents a 2-hours long conference to the Moscow highschool children at the Moscow State (Lomonosov) University MGU, read there the 13-th November 2004. It described some astonishing recent discoveries of the relations of the Galois elds to dynamical systems ergodic theory, to statistics and chaos, and also to the geometry of the projective structures on nite sets. Most of these recent discoveries summarized empirical studies, and some of the conjectures, provided by these numerical experiences, are still unproved, in spite of their simple statements, quite accesible to the highschool children (who might sudy them empirically, being virtually assisted by the computers). Together with these empirical studies continuation, it would be nice to investigate some remaining theoretical questions, like the natural problem of the projective permutations intrinsic characterization among all the permutations of a nite set: one should understand, which geometrical features of some special permutations of dozen points are making these special permutations projective, distinguishing them from the nonprojective permutations. These researches had been partially supported by RFBR, grant 05-01-00104. Keywords : Mathematics, algebra, geometry, number theory, Galois elds, dynamical systems, ergodic theory, mathematical statistics, projective geometry, projective line, Frobenius transformation, chaoticity, randomness, equipartition, primes, Euler group, Euler function, Cesaro averaging, geometrical progression, Fermat-Euler congruences, little Fermat theorem, binomial and multinomial coecients, Girard-Newton formula.

Contents1 What is a Galois eld? 2 Galois elds organization and table 3 Chaoticity and randomness of Galois eld tables numbers 5 13 21

4 Equipartition of geometrical progressions along a nite onedimensional torus 29 5 Adiabatic study of distribution of geometrical progressions of residues 43 6 Projective structures, generated by a Galois eld 51

7 Calculation of projective structures of nite projective lines, generated by elds of p2 elements, and of Frobenius transformations group action on these lines and on these structures 63 8 Cubic tables of elds 79

3

4

CONTENTS

Chapter 1 What is a Galois eld?A Galois eld is a eld, having a nite number of elements. Such elds belong to the small quantity of the most fundamental mathematical objects, serving to describe all other mathematical structures and models. The examples of such fundamental objects are the well known prime numbers, p = 2, 3, 5, 7, 11, 13, 17, 19, 23, . . . , 997, 1009, . . . , these are the positive integers, which have each only 2 integer divisors (1 and itself), the number 1 being not prime. The immediate natural science question, to which this notion leads, is already rather dicult: is the set of all the primes nite? (that is, whether the above sequence of primes might be continued indenitely?) The answer to this question had been discovered several thousands years ago: the prime numbers sequence is innite, a maximal prime number does not exist. To prove it, it suces to consider the number (2 3 5 p) + 1 , which provides the residue 1 while we divide it by any prime number 2, 3, . . . , p. This number is not, therefore, divisible by any of them. Hence it has a prime divisor, which is greater, than p. Therefore no maximal prime number p may exist. This remarkable mathematical reasoning is rather avoiding the question of highest interest from the natural sciences view-point: how often are primes encountered in the sequence of all the natural numbers {1, 2, 3, 4, 5, 6, . . . }? 5

6

What is a Galois eld?

Are the intervals between the consecutive prime numbers growing (while the numbers we consider become large)? What is the decimals number of the millionth prime? The rst natural scientist, studying this problem, has been Adrien Marie Legendre (1752-1833), who had considered (in the XVIII Century) the tables of the primes up to 106 and who had discovered empirically the following density decline of the primes distribution law: the average distance between the consecutive prime numbers, of order of n, grows with n like ln n (the natural logarithm), that is the like logarithm with base e 2, 71828 . . . , the Euler number e being e = lim 1 1+ kk

k

=

1 . m! m=0

For instance, ln 10 2, 3, and the average distance between the consecutive primes close to 10 is slightly greater than 2, since 7 5 = 2 , 11 7 = 4 , 13 11 = 2 . The primes in the region of n = 100 are 89, 97, 101, 103, their average 2 distance being therefore 4 3 . This distance should be compared to ln 100 = 2 ln 10 4, 6 of the Legendre law, and it is thus conrmed satisfactory already for n = 100. Of course, the existence of the pairs of the twins (that is, of the prime pairs, whose dierence is 2, like for 5 and 7, 17 and 19, 29 and 31) contradicts the growing of the distances between the consecutive prime numbers (provided that the twins number is innite, which is conjecturally true: this innity is one of the most celebrated unproved conjectures of the modern number theory). Unfortunately, the Legendre empirical observations had not been appreciated by the mathematical community of his time, since he had proving nothing, considering only some millions of examples. It is true, that he succeeded to deduce from his empirical statistical observations his law, being unable to provide the strict mathematical proof of the limit asymptotical coincidence of the averages of the distances with his proposed value ln n for n . Kolmogorov told me several times on his hydrodynamical turbulence studies: do not try to nd in my works any theorem, proving my statements: I

Chapter 1

7

am unable to deduce these statements from the basic (Navier-Stokes) equations of this theory. My results on the solutions of these equations are not proved, they are true, which is more important, than all proofs. The rst who appreciated the Legendre discoveries had been the Russian mathematician Tchebyshev. He proved rst, that even if the averaged distance between the consecutive primes in the neighbourhood of a large number n does not behave asymptotically as ln n, its relation to this Legendre suggestion remains bounded, the average distance belonging to the interval between c1 ln n and c2 ln n (where c1 < c2 were explicitly calculated). Later he proved more: provided that the oscillations between the above limits would extinct while n grows, leading for the average distance to the asymptotics c ln n with some constant c, then the constant c can not be different from 1. This is yet unsucient to prove the Legendre asymptotical formula, since there remains the possibility of the nonextincting oscillations between c1 ln n and c2 ln n, never leading to the c ln n behavior. However later (about 100 years after the Legendre discovery) two celebrated mathematicians, Hadamard (from France) and de la Valle Poussin e (from Belgium) proved, that the oscillations are indeed extincting for n , providing some c ln n asymptotical behaviour of the averaged distance between the consecutive primes in the neighbourhood of n. The world mathematical community claims therefore, that Hadamard and de la Valle Poussin made a great discovery (of the statistics of the large e prime numbers distribution). It seems to me, that this claim is rather unfair. These great mathematicians had only proved the existence of the distribution law (the existence of the constant c, unknown to them). Both facts of the natural science (the asymptotical proportionality of the average distance to ln n and the fact that the proportionality coecient equals 1) were discovered by Legendre and Tchebyshev, to whom one should attribute the great discovery of the primes density statistics, described above. In this high-school children conference I shall follow rather Legendre, than Hadamard: I shall talk on the empirical numerical observations, suggesting some new (and astonishing) laws of nature, whose transformation to the mathematical theorems state might wait some hundred years (as it had happened to the primes distribution law), in spite of the fact, that the discoveries of these new laws are quite accessible to the schoolchildren (even using no computers, while the computerized experiments might accelerate

8


the empirical experiences1 ). Besides the prime numbers, another example of fundamental mathematical objects is provided by the regular polyhedra (called also Platonic polyhedra, since they had been discovered by others). There are ve bodies: the tetrahedron (with 4 faces), the octahedron (with 8 faces), the cube (with 6 faces), the icosahedron (from the Greek icos for its 20 faces) and the dodecahedron (from the Greek dodeca for its 12 faces) see gure 1.1.

Tetrahedron

Octahedron

Cube

Icosahedron

Dodecahedron

Figure 1.1: Regular polyhedra. The dodecahedron had been used by Kepler to describe the distribution of the planetary orbits radii in the Solar system. The regular polyhedra are strangely related to a domain of Physics which seems to be quite dierent to the theory of the optical caustics, which provides, for instance, rst the explanation of the rainbow phenomenon (the rainbow angular radius being = 42 ) and second the theory of galaxies concentration in the Universe large scale structure. Kolmogorov explained, that the special beauty of the mathematical theories is due to the unexpected relations between quite dierent natural phenomena (say, between the theories of the electric and magnetic elds, provided by the Maxwell equations), which relations Mathematics discover. Unlike for the fundamental objects of the examples above, the Galois elds applications to the natural sciences are yet to be discovered. I hope, that they will appear rather soon, and I would like to shorten the remaining waiting time by my geometric presentation of the Galois elds theory. My description is closer to the natural science approach, than to the axiomaticoIn my personal experiments, leading me to the results below, no computers had been used, and my students, who had veried that the computerized experiments provide the same answers, discovered that my calculations contained several times less mistakes than that of the computers.1

Chapter 1

9

Figure 1.2: The rainbow origin. algebraic superabstraction style, dominating the existent presentations of this algebraic theory. The simplest example of a Galois eld is the residues eld modulo a prime number p (gure 1.3).1 2 0

3

4

Figure 1.3: A nite circle: Galois eld Z5 . Thus, for p = 2 we get the eld, consisting of two elements: Z2 = {0, 1} , with its usual arithmetics 0+0=0 , 0+1=1+0=1 , 1+1=0 ,

00= 01= 10 =0 ,

11=1 .

This binary arithmetics is the base of the computers, acting in the binary system. Thus, the simplest Galois eld is extremely useful: (the eld Z2 ) = (computers) .

10


The general eld notion is very similar to the preceeding example: there are two operations (called addition and multiplication), having the usual properties of commutativity, associativity and verifying the ordinary distributivity law; and one can divide the elds elements by every element of the eld, dierent from 0. The residues of the division by 3 form the eld Z3 , consisting of 3 elements {0, 1, 2} (where 1/2 = 2, since 2 2 = 1 for the residues modulo 3: (3a + 2)(3b + 2) = 9ab + 6a + 6b + 4 = 3c + 1). Very dierently, the 4 residues for the division of the integers by 4 do not form a eld, since the element 2 can not be inverted (the residue 2x is sometimes 0 sometimes 2, but it is dierent from 1, whatever residue would be x). However, there does exist a eld of 4 elements (the operations being dierent from the residues arithmetics modulo 4). To nd these operations is a useful exercise, which is neither too dicult, nor too easy for a beginner. The nite elds are called Galois elds, since he had discovered the following two remarkable properties of these elds: 1. The elements number of a nite eld is an integer of the form pn , where p is a prime; and for any prime p and any natural n there exists a nite eld having just pn elements. Thus, there exist elds with 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27 elements, but there does not exist any eld whose elements number is 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26 . 2. The eld, having pn elements, is dened by the number of its elements unambiguously (up to a elds isomorphism). Thus, a computer using the eld Z2 at Moscow, and another computer, working in Paris, might use two dierent copies of this eld. Say, one might denote the Paris eld elements and (instead of 0 and 1), dening the operations by the table +=+ = , = , + =+= ,

= = = .

Chapter 1

11

But this eld is isomorphic to the standard residues eld Z2 (diering only in the notations, 1 and 0). The notations independence of the content of the phenomena is the base of the relativity theory and of the whole relativistic physics. I shall not write here the proofs of the existence and of the uniqueness theorems for the eld of pn elements, formulated above. I shall describe instead the operations in this eld by explicit tables. Strangely, I had not seen in the published form the natural-science oriented description of the nite elds, presented below. Every eld contains the 0 element (zero), which does not change any element, to which it is added. All the other elements of the eld form the multiplicative group of the eld (being a group for the multiplication operation), since each nonzero element can be inverted. This group is always cyclic: there exists such an element A of the eld, that every non-zero element of the eld has the form Ak (where 1 k z 1 for the eld of z = pn elements). I shall not prove the cyclicity theorem (while its proof is not too dicult), since this theorem adds to the theory, described below, only the following axiomophilic addition: in the nature there are no other nite elds, dierent from the elds with a cyclic multiplicative subgroup. In other words, we might consider the theory, explained below, to describe the nite elds with an additional axiom (that the elds multiplicative group is cyclic) the existence of the primitive element A, whose powers provide all the nonzero elements of the eld. The constructions and results, explained below, belong to the theory of such elds. The absence of any dierent nite eld is a (nice) addition to this theory, but the theory itself does not depend on this additional property of our axioms. It is interesting to observe, that just the exagereated attention to the difcult studies of the axioms independence makes the algebraic and abstract theories of the mathematicians unnecessarily dicult and hostile for the natural scientists. Thus, the Lobachevsky plane is simply the unit circle interior disc, whose interior points are called Lobachevsky points, and whose Lobachevsky lines are the unit circle chords. The boundary circle (which does not belong to the Lobachevsky plane) is called the absolute. It is very easy to see, that these objects (forming the so-called F. Klein

12


model of the Lobachevsky plane since they had been invented by A. Cayley) verify all the Euclidean geometry axioms (there exists one and only one line, connecting two given points, etc.), except the parallels axiom: there exist an innity of Lobachevsky lines going through a given Lobachevsky point and having no common Lobachevsky points with a given Lobachevsky line, disjoint with the given Lobachevsky point (that is, an innity of chords, see gure 1.4).

Figure 1.4: Lobachevsky plane. These (obvious) natural science facts can be completed by a (dicult) theorem of the axiomophils: there exists no other Lobachevsky plane, dierent from the Klein model, described above (of course up to isomorphisms: the theorem states, that the Lobachevsky plane axioms imply the isomorphism of this plane to that of the Klein model). It is interesting, that Lobachevsky was unable to prove his main natural science belonging (and quite remarkable) statement: the parallelism axiom of the Euclidean geometry is independent from the other axioms (that is, it can not be deduced from them). The model, described above (and invented many years later, than Lobachevsky worked) proved just this independence statement. Indeed, if the wrongness of the Euclidean parallels axiom would imply a contradiction (which contradiction would be just the axioms proof), then the model would be contradictory too, providing therefore contradiction inside the usual Euclidean geometry (concerning the ordinary geometry of the chords of one circle). The proofs of the fundamental mathematical facts are in many cases much simpler, than the axiomophilic details, making so dicult the mathematical textbooks.

Chapter 2 Galois elds organization and tableThe multiplication in the Galois eld, consisting of n elements, 0 and {Ak }, 1 k n 1 is simply the addition of the logarithms k of the elements (considering these logarithms as the residues of the numbers k modulo n1): 0 Ak = 0 , Ak A = Ak+

(if k + > n 1, one replaces the sum by k + (n 1) to reduce the sum to a value, smaller than n). It remains to dene the addition operation. Denoting the elds element k A by the sign k, we arrive to the following tropical operation over these logarithms: Ak + A = Ak . The modern term tropical, meaning exotic, is used when one lowers the level of the algebraic operations, transforming the multiplication to the addition, replacing the addition by the lower level tropical addition operation, with respect to which the usual addition is as distributive, as is the usual multiplication with respect to the usual addition: x(y + z) = xy + xz , x + (y z) = (x + y) (x + z) .

An example of such a tropical addition is the operation x y = max(x, y) for the real numbers. One would be able to obtain this tropical operation from the usual addition using the logarithms accompanied by the short waves 13

14

Galois elds table.

asymptotical expansion of quantum mechanics, where the wave length h is approaching 0. The relation x h y = ln(ex/h + ey/h ) h denes the tropical addition operation h , tending to max(x, y) for h 0. While all these things are obvious, they imply a nonobvious tropical conclusion: replacing the multiplication and the addition operations with their tropical versions (addition and maximum), one can transform many formulae and theorems of the calculus (like the Fourier series theory) into its (nonevident) tropical versions, providing interesting results in the convex calculus and linear programming theories. Consider for simplicity the case of the eld F of z = p2 elements. It contains the scalar elements 1, 2 = 1 + 1, . . . . The eld being nite, one of the sums would coincide with the other. Hence some sum of m ones (equal to the dierence of the coincident sums) equals to zero, m = 1 + + 1 = 0. We shall suppose the repetitions number m to be the minimal value, providing the 0 scalar. We shall prove now, that m = p. Indeed, call any element x equivalent to any element x + 1 + + 1 (where the number of the ones is at most m). Each equivalence class consists from m elements, and these classes are disjoint. Therefore, the scalar elements number, m, is a divisor of the number p2 of the elements of the eld. Thus m is either p or p2 . The second case is impossible. Indeed, consider the scalar element x = 1 + + 1 (p times). This element of the eld having p2 elements has no inverse element, since no integer of the form pq provides the residue 1 while one divides it by p2 . Therefore, x = 0, and the scalars number is m = p. Consider the element 1 together with the primitive element A of our eld. Adding each of them less than p times, we create the p2 sums uA + v1. All these elements of the eld are dierent (otherwise we would obtain A = (v/u) 1, and therefore all the elements of the eld would be scalars, which is impossible, since the number of the scalars is p, which is smaller than p2 ). Thus the eld of p2 elements consists exactly of the linear combinations F = {uA + v1} with coecients u Zp , v Zp . In this sense we had distributed all the elements of the eld in the cases of a p p square (or rather of the nite torus Z2 of gure 2.1, being the p 2-plane over the eld Zp ).

Chapter 2

15

Therefore, we lled the z = p2 cells of this nite torus by the p2 logarithmical symbols {; 1, . . . , z 1} (where the symbol k, which is a residue modulo z 1, denotes the element Ak of the eld F , the symbol representing1 the elds zero element).R2

Figure 2.1: The continuous torus and the nite torus, consisting of 4 points. This lling provides the tropical operation simple description. Namely, the sum of the elds elements corresponding to the symbols k and , Ak = u A + v 1 , A =u A+v 1 ,

is (u + u )A + (v + v )1 = Ak . Therefore (gure 2.2), the symbol k lls in the table of the eld the vector sum of the places of the symbols k and : the addition operation of the eld F (consisting of p2 elements) is represented by the vector addition of the places of the added elements in the table of the eld.u k k

v

Figure 2.2: Tropical addition of the numbers k and in the table of the eld. Thus, to describe the eld of z = p2 elements it suces to calculate the places (uk , vk ) of the elements Ak = uk A + v k 11

(1 k z 1)

The schoolchildren suggested at the lecture to denote ln 0 by , but I leave the symbol , being unaware, whether A > 1 in F .

16

Galois elds table.

in the table of the eld. This calculation is an (easy) extension of the Fibonacci numbers ak recurrent construction (these numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . , ak+2 = ak+1 + ak , describe the rabbits population growth). Namely, suppose that in eld F we have A2 = A + 1 , Then we nd in F the relation A3 = A(A + 1) = (A + 1) + A = (2 + )A + 1 . Continuing in this way, we get the recurrent relation, expressing recurrently the places of the elements Ak in the table of the eld, uk+1 = uk + vk , vk+1 = vk . (2.2) ( Zp , Zp ) . (2.1)

Therefore, the 2 residues and (modulo p) provide consecutively the places (uk , vk ) of all the elements Ak in the table of the eld. To obtain the table, it remains to choose the values of the parameters 2 and . One should choose them in such a way, that, rst, Ap 1 = 1, (that is, up2 1 = 0, vp2 1 = 1) and, secondly, all the preceding vectors (uk , vk ), (1 k < p2 1) should be dierent from the vector (0, 1). In principle, one might try in turn all the p2 pairs of residues (, ) to choose the convinient parameters values. The number of trials is even not too large. For instance, if p = 5 both conditions above are fullled by the pair = = 2. However, one might seriously accelerate the choice of the parameters values, using the Pascal triangle of the binomial coecients. Namely, it is easy to prove the following explicit formula for the recurrent relations (2.2) uk =t s t Cs+t ,

(2.3)

where the powers s and t are related by the homogeneity condition of the coecient uk in A, implied by the condition (2.1). This condition provides the weights (deg = 1, deg = 2), implying for deg uk = k 1 the homogeneity relation s + 2t = k 1 for the degrees s and t of the monomials of the formula (2.3) for the quantity uk . For instance, for k = 6, the Pascal triangle provides the following coecients of formula (2.3):

Chapter 2

17

1 1 1 1 1 1 5 4 10 3 6 10 2 3 4 5 1 1 1 1 1 u6

Therefore, the quantity u6 is represented as the sum of three monomials of weight 5: u6 = 3 2 + 43 + 15 . Using this algorithm, I calculated the places of the 24 nonzero elements A of the eld, consisting of 25 elements in half an hour. The resulting table of this eld isk

p=5

u 4 3 2 1 0

13 7 19 1 0

15 10 11 8 24 1

5 9 2 4 18 2

16 14 21 17 6 3

20 23 22 3 12 4 v

Example. A10 = 3 A + 1 1, A19 + A8 = A10 . Remark. The symmetry center, denoted by the sign has the following property (easy to prove): k = 12 (mod 24) whenever the symbols k and places are situated symmetrically with respect to this center (on the nite torus). For instance, 21 9 = 12, 17 5 = 12, 24 12 = 12 (it would be equal to (z 1)/2 for the eld of z elements). The reason of this symmetry is the evident identity A12 = 1 (that is, u12 = 0, v12 = 4). The symmetry allows one to reduce the elds table calculation, making it two times faster: it suces to nd the coordinates (uk , vk ) of the symbols k z/2 for the case of the eld having z = pn elements.

18

Galois elds table.

The elds table may be interpreted the following way. The multiplication of the elements of the eld by A acts on Ak as a linear operator on the plane of the table: Ak 1 = u k A + v k 1 , Therefore, the matrix of this linear operator on the plane with coordinates u and v, equipped with the basis (1, A), has the form (Ak ) = vk vk+1 uk uk+1 . Ak A = uk+1 A + vk+1 1 .

For k = 1, this matrix is equal to (A) = 0 1 equal to 0 2 1 2 for p = 5 .

The relation (2.1) is simply the characteristic equation for matrix (A). The operator of the multiplication by Ak being the k-th power of the multiplication by A, the matrix (Ak ) is the k-th power of the matrix (A). Therefore, the elds table construction provides a representation of the eld, consisting of p2 elements, by the second order matrices (Ak ), whose elements belong to Zp (being residues for the division by p). The elds operations are represented as the matrices addition and multiplication (A)k (A) = (A)k+ , (A)k + (A) = (A)k . For the eld, consisting of z = pn elements, a similar construction provides a eld representation by the matrices of order n with elements in the eld Zp . Some examples, where n = 3, are listed below, in 8. For the elds of p2 elements, where p = 7, 11 end 13 the calculations, quite similar to those described above for p = 5, provide the following answers: p=7 (A) = 0 2 1 2 p = 11 (A) = 0 3 1 1 p = 13 (A) = 0 2 1 4 .

The resulting table of the eld of p2 = 49 elements lls the nite torus Z2 7 by the following residues (modulo 48):

Chapter 2u 6 5 4 3 2 1 0 25 9 17 41 33 1 0 44 35 37 23 38 18 48 1 7 30 28 29 22 36 19 10 2 15 21 27 32 40 2 3 3 39 34 12 5 6 16 4 45 42 26 14 43 47 46 13 4 11 31 20 8 24 5 6 v

19

p=7

The table of the eld of p2 = 121 elements lls the nite torus Z2 by the 11 following residues (modulo 120):

u10 61 11 15 76 22 43 78 53 62 56 105 9 25 42 95 17 99 26 40 20 106 69 87 30 57 28 7 5 8 13 94 14 83 115 8 7 109 4

6 104 59 70 101 33 63 91 110

p = 11

6 37 32 29 19 52 107 81 38 54 118 111 5 97 51 58 114 98 21 47 112 79 89 92 4 49 50 31 3 2 1 0 35 67 9 1 0 93 41 10 119 44 66 64 3 73 65 88 117 90 27 68 55 23 74 34 46 80 100 86 39 77 35 102 45 116 2 113 18 103 82 16 75 71 1 2 3 4 5 6 7 8 9 10

120 84 72 48 96 36 108 12 24 60

v

Bold numbers in these tables represent the multiplicative generators Ak of the multiplicative groups of the elds (corresponding to these values of k, which are relatively prime to z 1 = p2 1 = 120 in our case p = 11). The table of the eld of p2 = 169 elements lls the nite torus Z2 by the 13 following residues (modulo 168 = 23 3 7):

20

Galois elds table.

u12 35 45 161 165 13 76 58 47 158 122 23 166 64 11 15 145 143 88 91 52 95 121 111 96 6 162 156 10 141 10 132 101 79 114 49 54 103 53 120 46 69 9 113 51 75 41 73 26 18 104 21 92 150 86 25 8 127 32 39 40 100 87 164 55 106 89 35 65 118 7 71 152 108 33 62 151 31 50 5 43 34 149 119 5 4 29 109 2 3 2 1 0 66 22 139 80 9 144 44 167 147 3 16 124 123 116

p = 13

6 155 63 83 128 60 93 134 115 67 146 117 24 68 8 105 20 102 110 157 125 159 135 4 59 61

57 153 130 36 137 19 138 133 30 163 17 48 94 99 72 78 90 12 27 37 11 136 7 1 148 82 107 38 74 131 142 160 97 81 77 129 168 98 56 28 42 154 70 126 112 140 14 84 0 1 2 3 4 5 6 7 8 9 10 11 12

v

Remark. While the eld is dened unambiguously by its elements number, the table of this eld is not dened by it, being dependent of the choice of the multiplicative generator A of the group of the nonzero elements of the eld. Instead of the generator A, one might choose a dierent primitive element, A = Ak (which is primitive just when k is relatively prime to the number z 1 for a eld of z elements). The primitive elements form the Euler group (z 1). We discuss the inuence of the choice of the primitive element A on the constructions described below in 7: these investigations lead to some astonishing facts of the projective geometry of nite sets (see 6 and 7 below).

Chapter 3 Chaoticity and randomness of Galois eld tables numbersLooking at the preceding tables of Galois elds, one has the impression, that their llings by the integers from 1 to z 1 (z being the elds elements number, equal to p2 in our examples) behave like some kind of random numbers tables: it is dicult to guess the place of the next symbol k + 1, knowing the preceding symbols k place. The attempts to formulate this natural science observation as a mathematical statement lead to hundreds of conjectures. Perhaps, most of these conjectures will become interesting proved theorems in the future (while at present it had happened only to few of these conjectures). The general scheme of the randomness conjectures formulations is the following reasoning. The genuine random llings have several properties, studied in probability theory and in the stochastic processes theory. To check the randomness of the lling of the table of the eld, choose one of these properties, and verify, whether the quasirandom numbers, lling the table, approximatively do verify it. The conjecture, to which one arrives this way, claims, that the randomness criterium, that one had chosen, is approximatively fullled by the matrix of the eld of z = pn elements better and better, while the prime number p (for a xed n) or the elements number z is growing. The limiting relation (for p ) is conjectured to be fullled exactly. Thus, to x the mathematical formulation of the pseudo-randomness conjecture one has to describe exactly the chosen criterium. Since there exist 21

22

Chaoticity of elds table.

many such criteria, one gets many conjectures. I shall discuss below a short list of simplest examples (which are already nontrivial and interesting, in spite of their simplicity: every highschool student might check these conjectures empirically, xing p, even using no computers help). Let us start from an example: suppose, that the whole table is subdivided into two disjoint parts, and count those numbers k, lling the table, which occur in the rst part, G. For a random lling the number N of the occurrences of G among m trials is proportional to the volume (to the area in the 2-dimensional case) of the domain G: N |G| , m z (where z is the total volume of the table, that is z = p2 in the case of the eld of p2 elements). Of course, for m = z the approximated equality, written above, is exactly fullled by the sequence of the elds elements Ak , 1 k m, since each cell of the table, belonging to the domain G (supposed to consist of tables cells) is visited by the sequence {Ak } just once. Therefore, to formulate a nontrivial conjecture on the equidistribution of the elements {Ak } along the eld one should take only a part of the whole sequence. Choose for this a number strictly between 0 and 1 and consider the beginning of the sequence {Ak }, consisting of m z members (1 k m). Then we get the following Geometrical progression equidistribution conjecture for the Galois eld of z = pn elements: |G| N = , (3.1) lim p m z where N is the number of those rst m elements of the sequence {Ak } of elements of the eld of z elements, which do belong to the domain G. I had xed here the tables dimension n, but one might consider also the z limits, where n is not xed. Example. For the eld of z = p2 = 25 elements, choose as G the union of the rst two columns of the table: |G| = 10, v = 0 or 1 in G. Let us take the rst half of the geometrical progression {Ak }, 1 k (12 = m). Then the visits number (counted by the above table of the eld, page 17) is N = 5. The deviation from the theoretical randomness criterium (3.1) is

Chapter 3 in this case

23

N |G| 5 10 1 = = . m z 12 25 60 Therefore, even while the prime number p = 5 is not so large, the approximation to the equidistribution is rather good. I had not proved the limit theorem (3.1) in its general form, but I discuss below (in 4) some of its versions, which are proved. As other randomness criteria one might suggest, for instance, the following ones. Subdivide the eld in two disjoint domains F =GH , whose elements numbers are, correspondingly, |F | = z , |G| = rz , |H| = sz ,

where r + s = 1. For a genuinely random sequence of the choices of the elds elements Ak in F the frequencies of the jumps from G to G, from G to H, from H to G and from H to H for the transition to the next element of the sequence are, correspondingly, r 2 , rs, sr and s2 . For the geometrical progression {Ak } (which one might leave unshortened in this case: 1 k < z) one expects similar frequencies of the four events (Ak G, Ak+ G), (Ak G, Ak+ H) and so on. Geometrical progression {Ak } mixing conjecture for the z = pn elements eld. The numbers N (G, G), N (G, H), N (H, G) and N (H, H) of the occurrences of the k, for which (Ak G, Ak+ G), (Ak G, Ak+ H), and so on are asymptotically proportional to the frequencies (r 2 , rs, sr, s2 ) of the random jumps:p

lim

N (G, G) = r2 , z

p

lim

N (G, H) = rs , z

...

.

One more randomness criterium is provided by the table variation, which measures the dierences of the symbols k at the neighbouring cells of the

24


table. For instance, one might counts the sum of the dierences |k |, or better the sum of the distances (k, ) of the neighbouring symbols in the table, comparing it with the similar sum for a purely random lling of the table by the symbols k from 1 to z. The asymptotical dierence of every of these 2 sums for the eld (which dier) from their mathematical expectation for a random lling is expected to reach (relatively) small values for large primes p (or for the growing numbers z = pn of elements of the elds). For the table of the eld having p2 = 25 elements (at page 21), the observed averaged distance (k, l) between the residues k, l occupying the neighbouring places in the toric table, is 6.41. The random lling expectation of this distance (between the residues modulo p2 1) is (p2 1)/4 = 6. For the table of the eld, having p2 = 169 elements (at page 25) the observed averaged distance (k, l) between the residues k, l, horizontal neighbouring in the table, equals 42.0299, the random lling expectation being (p2 1)/4 = 42. The neighbouring in the table is to be considered here accordingly to the torical geometry (say, for p = 5 the value u = 4 is a neighboor of u = 0( 5)). In a similar way, one might consider a dierent kind variation, measuring the distances of the places of the symbols k and k +1 in the table. One might consider the quantity = k (k, k + 1), summing either for the cyclically closed sequence, or for the usual sequence, 1 k < z. The averaged distance (k, k + 1) between the places of the neighbouring residues in the toric p p table of the eld, having p2 = 25 elements, is 13 observed to be 2 24 , the random lling providing the expected average p/2 = 1 2 2 . The distance between the places was measured using the sum of the dierence of the coordinates (using the torus geometry, that is, considering the coordinates as being residues modulo p). Counting the summands of this sum one should not forget the torical geometry of the table, lling Zn by the symbols k. For instance, for p = 7 p and n = 1 consider the lling of the 7 consecutive places of the table by the cyclical sequence of values k = (1, 5, 4, 2, 3, 7, 6) (gure 3.1). For this lling one gets = 3 + 1 + 2 + 1 + 2 + 1 + 2 = 11 , = 4 + 1 + 2 + 1 + 3 + 1 + 2 = 14 , since (3, 7) = 3 and (6, 1) = 2 in the torical geometry of Z7 .

Chapter 32 3 1 4 7 5 6

25

Figure 3.1: Distance 2 between points of a nite torus. The variation takes the value 11 on this cyclical sequence of the seven residues modulo 7. For these variations the elds table randomness conjecture suggests the relatively small dierence between the quantity , calculated from the table of z = pn elements eld, and the mathematical expectation of a similar sum for a genuinely random lling of the table, provided that p (or z) is suciently large. One might use here either the sum of the m distances between the m elements of a cyclical sequence, or the sum of the m 1 distances between the elements of an ordinary sequence of m elements. As one more randomness characteristics of the set {Ak } in the table one might use such quantity, as the minimal radius r(m) of the balls, centered at the rst m points of the set, which cover together all the table, or one might consider the maximal radius R(m) of the ball, containing no points of this subset (gure 3.2).

R r

Figure 3.2: Covering balls and void balls of a set of points One should compare the values r(m) and R(m), calculated from the elds tables, with the similar characteristics of m genuinely random points: the eld table chaoticity conjecture claims the similarity of these quantities behaviours

26


for the elds of z = pn elements (where p or z ), provided that m z (where 0 < < 1 is xed). One more randomness characteristic of a set of m points of the table is the percolation radius, dened the following way. Enclose each point of the set in a radius r ball, centered at this point. If r is suciently small, one cant cross the table from one side to the opposite one (creating an uncontractable path on the torus) along these small balls union. If the radius is suciently large, such a percolation through the union of the balls, representing the defects of the material, becomes possible (the word percolation is borrowed from the leaks studies in the materials of the vessels) see gure 3.3.

no percolation

percolation

Figure 3.3: Percolation appearence, due to the defects radius growth. The critical (minimal) value r(m), rst providing the percolation apparence, is called the percolation radius of a given set of m points: it is the smallest radius of the defects, producing the leakage. The percolation chaoticity conjecture for the points of the geometrical progression {Ak } in the elds table compares the percolation radius behaviour r(m) for m independent random points of the table (say, for m z, where 0 < < 1 is xed and the eld contains a large number z = pn of elements). Here, as above, I mean the limiting behaviour for p (but the z limit might also be considered). In these percolation studies it might be also interesting to replace the balls of radius r by the segments of the progression {Ak : |k k0 | } , comparing it with a similar set of segments of a random sequence {Ak } of m points of the elds table. Dening this way the quasi-percolation radii (m), their conjectured behaviours should be similar for the rst m z points

Chapter 3

27

{Ak } of the z = pn elements elds table and for the random sequence of m points of the table (where, as usually, 0 < < 1 is xed and p (either z )). Of course, one is able to invent many dierent criteria of the table chaoticity, and every one of them leads to an (interesting?) conjecture of ergodic character, which deserves to be studied empirically (and which would, hopefully, become a theorem, if the numerical experiences would conrm it, which theorem one might later try to prove). The resulting theory is some number-theoretical nitistic version of the ergodic theory of the tori automorphisms, where the chaoticity and the mixing properties of the progressions {Ak } had been studied for the volumes preserving automorphisms A of the continuous torus T n . The dierence of our case depends on the fact, that the nite torus Zn p consists of a nite number z of points, and that the innite time limit, used in the ergodic theory to dene the time average, is replaced in our case by the limit for the growing number m z of the points in the orbit of the dynamical system which we are studying (occurring due to the growing of the parameter p or of the number z = pn of the points of the nite torus). It is interesting, that the percolation chaoticity problem had not been studied, as far as I know, even in the (more simple) case of the continuous torus hyperbolic automorphisms of ergodic theory.

28


Chapter 4 Equipartition of geometrical progressions along a nite one-dimensional torusThere exists two very dierent ways to formulate a problem: the French way consists in the most general formulation, leaving no possibilities for further generalizations (diering from the nonsense), the opposite Russian way is to choose the simplest case which can not be simplied further (preserving some content of the problem)1 . I had tried above to formulate the elds tables randomness conjectures in the French form. Let us consider now the Russian form of the rst of these conjectures, claiming the equidistribution of the progression in the eld of pn elements. To do it, we restrict ourselves, considering the simplest eld Zp (consisting of the p residues of the division by a prime number p), that is consider the simplest case n = 1 of the general theory of 3. To simplify the formulae, we shall suppose the prime number p to be odd, and as the domain G of 3 we shall chose the rst half of the nonzero elements of the eld, {c : 1 c (p 1)/2}, |G| = (p 1)/2. As the segment of the geometrical progression of residues we consider its rst m = (p 3)/2 terms, {Ak : 1 k m}.Tchebyshev, who had a lot of friendly relations to French mathematicians, like Liouville, had never discussed with them any mathematics, to make no harm to his Russian approach by their inuence, as he described it, returning home.1

29

30

Geometric progressions equipartition.

This strange choice of the half of the progression ( 1/2) is explained by the little Fermat theorem statement, Ap1 = 1, making Ak = 1 for k = (p 1)/2. Therefore this term of the progression is not random at all, while the randomness might be expected for smaller k, that is for k (p 3)/2. Calculating these segments of the progressions of residues modulo p for p = 5, 7, 11 and 13, we should rst nd for every p all the primitive elements A (for which the smallest period T of the progression takes just the Fermat theorem value T = p 1, while for other A it might be a smaller divisor of p 1). These progressions and periods for p = 5 are provided by the following table: A 1 2 3 4 {Ak } T 1, 1, 1, 1 1 2, 4, 3, 1 4 3, 4, 2, 1 4 4, 1, 4, 1 2 N 1 0 2+1+2+1=6 1+2+1+2=6

The primitive elements are here A = 2 and A = 3; they are denoted by the bold characters. The column N represents the number of the visits of the segment of the rst m = (p 3)/2 terms of the progression to the domain G (consisting of the residues, which do not exceed (p 1)/2). For p = 5 we obtain m = 1, (p 1)/2 = 2, and therefore N (A = 2) = 1 and N (A = 3) = 0. The column represents the sum of the distances in Z5 between the consecutive members of the progression (considering the progression as a cyclic sequence, that is including also the distance from the last member of the period to the rst one). The dierences between the observed frequencies and the space average (measuring the error of the equipartition conjecture for the geometric progressions residues in the space of the nonzero residues) are equal to: A=2 : A=3 : |G| 1 2 1 N = = , m z1 1 4 2

N |G| 0 2 1 = = , m z1 1 4 2 We observe that in both cases the error of the approximation to the equidistribution has the absolute value 1/2. In the average (with respect to

Chapter 4

31

the choice of the primitive element A) the equipartition criterium is fullled exactly |G| N = , m z1 where N= = N (A) /(the number of the primitive elements A) = (1 + 0) . 2

Similar calculations for the prime number p = 7 provide the following table of the answers (p = 7, m = 2, |G| = 3, z = 7): A 1 2 3 4 5 6 {Ak } 1, 1, . . . 2, 4, 1, . . . 3, 2, 6, 4, 5, 1, . . . 4, 2, 1, . . . 5, 4, 6, 2, 3, 1, . . . 6, 1, 6, 1, . . . T 1 3 6 3 6 2 N

2 0

1 + 3 + 2 + 1 + 3 + 2 = 12 1 + 2 + 3 + 1 + 2 + 3 = 12

(we used the fact, that the distance between the elements 1 and 5 of Z7 equals 3). Thus, the mean number of visits of G is equal to N = (2 + 0)/2 = 1, therefore, N /m = 1/2. The spatial average also equals to |G| 3 1 = = . z1 6 2 Therefore, the equidistribution criterion is once more fullled exactly (in the average with respect to the primitive element A choice), like for p = 5. The answers for the case p = 11 take the form p = z = 11, m = 4, |G| = 5, z 1 = 10,

32 A 1 2 3 4 5 6 7 8 9 10

Geometric progressions equipartition. {Ak } 1, 1, . . . 2, 4, 8, 5, 10, 9, 7, 3, 6, 1 3, 9, 5, 4, 1, . . . 4, 5, 9, 3, 1, . . . 5, 3, 4, 9, 1, . . . 6, 3, 7, 9, 10, 5, 8, 4, 2, 1 7, 5, 2, 3, 10, 4, 6, 9, 8, 1 8, 9, 6, 4, 10, 3, 2, 5, 7, 1 9, 4, 3, 5, 1, . . . 10, 1, 10, 1, . . . T 1 10 5 5 5 10 10 10 5 2 N 3 30

1 3 1

30 30 30

We see from this table, that the visits statistics provides the values

N=

3+1+3+1 =2, 4

1 N = , m 2

the space average being equal to

|G| 5 1 = = . z1 10 2 Thus the equipartition criterium is fullled exactly in the average under the choice of the primitive element A (the absolute values of the error for the particular choices of A being all equal to 1/4).

The answers for the case p = 13 provide the values z = 13, m = 5, |G| = 6, z 1 = 12. The progression table takes in the case p = 13 the form

Chapter 4 A 1 2 3 4 5 6 7 8 9 10 11 12 {Ak } 1, 1, . . . 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, . . . 3, 9, 1, . . . 4, 3, 12, 9, 3, 1, . . . 5, 12, 8, 1, . . . 6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1, . . . 7, 10, 5, 9, 11, 12, 6, 3, 8, 4, 2, 1, . . . 8, 12, 5, 1, . . . 9, 3, 1, . . . 10, 9, 12, 3, 4, 1, . . . 11, 4, 5, 3, 7, 12, 2, 9, 8, 10, 6, 1, . . . 12, 1, 12, 1, . . . T 4 12 3 6 4 12 12 4 3 6 12 2 N 4 42

33

2 1

42 42

3

42

In this case the visits number, averaged along the 4 primitive elements of the eld, equals 4+2+1+3 1 N= =2 , 4 2 therefore N /m = 1/2. The space average also takes the value |G| 6 1 = = . z1 12 2 Thus, for p = 13, the equipartition criterium is also fullled exactly (in the average with respect to the choice of the primitive element A). The individual choices (A = 2, 6, 7, 11) provide, correspondingly, the errors (3/10, 1/10, 3/10, 1/10) . These empirical studies lead us to the following conclusion. Theorem. The equipartition criterium for the distribution of the rst m = (p 3)/2 members of the progression {Ak } distribution among the nonzero residues of the division by p is exactly fullled (in the average with respect to the choice of the primitive element A) for the domain G = {1 c (p 1)/2 = |G|} in the eld Zp (for any odd prime number p).

34

Geometric progressions equipartition. In other words, for the averaged visits number N= N (A) , (number of the primitive elements A)

we have the ergodic value N |G| 1 = = . m p1 2 Proof. Together with a primitive element A the inverse residue modulo p, B = A1 , is also a primitive element. Lemma. The following identity holds: N (A) + N (B) = m . Proof of the Lemma. Taking into account the Fermat congruence Ap1 = A2m+2 = 1, we deduce that the two sequences {1 k m} and {1 m} of the progressions Ak and B = Ap1 cover with multiplicity one all the progression {Ai , 1 i p 1}, except its two trivial (nonrandom) terms, Ap1 = 1 and Am+1 = 1. Therefore they cover (with multiplicity 1) every element c of domain G, {2, 3, . . . , m + 1}, except the element c = 1. Thus the sum N (A) + N (B) of the visits numbers for both sequences to domain G equals m, and the Lemma is therefore proved. The Lemma implies the equality of the mean (over the choices of primitive element A) visits number N to m = (p 3)/2. Indeed, the whole set of the primitive residues A consists of (disjoint) pairs of the form {A, B}, where AB = 1 (as a residue modulo p). Each pair provides the contribution m to the sum N (A), accordingly to the Lemma. Therefore, the whole sum equals m, whence we nd the averaged number of visits, N (A) m m N= = = . 2 2 2 Thus, N /m = 1/2, which proves the Theorems statement.

Chapter 4

35

The above tables show, that in all our examples the variation = (Ak , Ak+1 ) of the whole progression of p1 residues, considered as points on the nite circle Zp , equals the value = p2 1 , 4

independently of the choice of the primitive residue A. The mean variation of a random cyclical sequence of p 1 = 4 points of Z5 can easily be calculated. It suces to consider the 6 sequences, starting with the residue modulo 1, (1, 2, 3, 4) , (1, 2, 4, 3) , (1, 3, 2, 4) , (1, 3, 4, 2) , (1, 4, 2, 3) , (1, 4, 3, 2) . Their variations are, correspondingly, equal to (5, 6, 7, 6, 7, 5) (we had used the distance (4, 1) = 2 in Z5 ). Therefore, the mean value of the variation of a cyclical sequence of 4 points on the nite circle Z5 is = 6. Thus, the variations of the cyclical geometrical progressions, formed by the powers of the primitive residues A (of the division by p = 5), calculated above, (A) = 6, coincide with the mean variation = 6 of a random cyclical sequence (of the same length p 1 = 4) of elements of Z5 . This observation provides one more argument for the quasirandomness statement (of the table of the eld of p = 5 elements). When the prime number p is growing, the mean variation of a random sequence of p 1 points on the nite circle Zp grows like p2 /4. This follows from the argument below. The distance between two randomly choosen points of this nite circle attains the values from 1 to (p1)/2. Its mean value is easily calculated to be close to p/4. Therefore, the sum of all such distances between the consecutive points of our sequence (there are p 1 such distances) is asymptotically growing with p like p2 /4 (with a declining relative error). Thus, the calculations of the variations of the cyclical geometrical progressions of lengths p 1 in the elds Zp (for p 13), presented above, conrm once more the quasirandomness of the table of the nite eld with p elements. Remark (on logarithms complexity). The chaoticity of the distribution of the geometric progression of the residues leads to interesting facts and conjectures of complexity theory. If a is a primitive residue modulo p, each

36


nonzero residue x modulo p has the form ak , and the complexity conjecture is, that to calculate the logarithm k of x is a dicult computational problem. To dene the diculty measure, one might classify functions ( on a nite set with a nite number of values) according to the degree of complexity of the formula, dening this function. To dene numbers, measuring complexity, I shall consider the simplest case of binary functions f : (Z/nZ) (Z/2Z). Such a function can be considered as a sequence (x1 , . . . , xn ) of n elements, each of them being either 0 or 1. There are 2n such sequences, and they form the modulo 2 vector-space (Z2 )n . One can consider these functions as the vertices of a cube of dimension n. To measure the complexity of a function x following Newtons idea, we associate to it the rst dierence function, y, dened as the sequence of the binary residues y(k) = x(k + 1) x(k) (mod 2) . (The argument k being a residue modulo n, we get n dierences of n residues, considering the next element after the last one to be the starting element of the sequence. Making the sequence cyclic, we avoid the boundary eects). Thus, if x = (1, 0, 0, 1, 1) we obtain the dierences y = (1, 0, 1, 0, 0) (since (y5 = x1 x5 = 0). The dierences operator is a linear operator (abelian group homomorphism) A : Zn Zn . 2 2

The complexity of a point x Zn will be calculated in terms of the 2 sequence of the consecutive dierences, At (x) Zn 2 (t = 1, 2, . . . ) .

Example. For the constant function x we get A(x) = At (x) = 0. For a polynomial x of degree d (that is, for x(k) = a0 k d + + ad ) we get At x = 0 for any t > d. We shall study below the spectral properties of the linear operator A. It is natural to consider the constants to be the simplest functions, the small degree polynomials to be less complicated, than the higher degree polynomials, and to consider the nonpolynomial functions as even more complicated objects. (I shall not formulate the evident next steps, involving the

Chapter 4

37

exponents and the dierential equations solutions the reader might construct his hierarchy of more and more complicate functions x according to his needs). The conjecture is, that the the logarithmic functions, dened above, are complicate. I shall not prove this conjecture, but I shall show several examples, conrming it. Other conjectures claim, that most of the 2n functions, forming Zn , are 2 like random sequences (at least asymptotically for n and at least for the majority of these functions). I shall not prove it, but the examples, discussed below, provide the complicate functions, behaving similarly to the random sequences in the numerical experiments. I hope, that this quasirandom behaviour is rather a general phenomenon, than a special property of our examples. To understand the complexity ranges, we start from the general study of the dierences operation A : Zn Zn . 2 2 Being a mapping of a nite set to itself, the operator A decomposes the set Zn into connected invariant components. 2 We consider these components as directed graphs (which edge leaving x connect point x with Ax). Each connected component of the graph of a mapping consists of an attracting cycle Om of some length m 1:

O1 =

, O2 =

, O3 =

, ... ,

and of the trees, attracted by the vertices of the cycle. We shall need the binary trees T2q of 2q vertices:

T2 =

, T4 =

, T8 =

, ... .

We shall denote by Om T2q the component, whose cycle Om is attracting a tree T2q at each vertex: O1 T2q = T2q , and next products are

O3 T2 =

,

O2 T 4 =

, ... .

38


Theorem 1. The graph of the dierences operator A : Zn Zn has the 2 2 form, presented in the following table, for n 12: n 2 3 4 5 6 7 8 9 10 11 12number of the components

cycles and trees (O1 T4 ) (O3 T2 ) + (O1 T2 ) (O1 T16 ) (O15 T2 ) + (O1 T2 ) 2(O6 T4 ) + (O3 T4 ) + (O1 T4 ) 9(O7 T4 ) + (O1 T2 ) (O1 T256 ) 4(O63 T2 ) + (O3 T2 ) + (O1 T2 ) 8(O30 T4 ) + (O15 T4 ) + (O1 T4 ) 3(O341 T2 ) + (O1 T2 ) 20(O12 T16 ) + 2(O6 T16 ) + (O3 T16 ) + (O1 T16 )

A u = Av A2 = 0 A4 = A A4 = 0 A16 = A A8 = A 2 A8 = A A8 = 0 A64 = A A32 = A2 A342 = A A16 = A4

1 2 1 2 4 10 1 6 10 4 24

The proof is a direct verication. Say, for n = 2 there are 2n = 4 vertices, and the dierences operator is, by denition, acting the following way: A(0, 0) = (0, 0) , A(0, 1) = (1, 1) , A(1, 0) = (1, 1) , A(1, 1) = (0, 0) , providing the graph, whose only component is

(0, 1) T4 : (1, 1) (1, 0) (0, 0)(4.1)

Denoting by the shift operator (x)k := xk+1 , we obtain the formulas A = 1 + , n = 1. Therefore, we get, for n = 3, A = 1 + , A2 = 1 + 2 + 2 = 1 + 2 , A3 = 1 + + 2 + 3 = + 2 A4 = + 2 + 2 + 3 = + 1 = A . Several evident properties of the formulas in the table are easily provable in the general situation. Thus, for n = 2m one has An = 0, since all the i binomial coecients Cn are even for i = 0, n: An = 1 + n = 1 + 1 = 0 (mod 2) .

Chapter 4

39

Now we shall study the places of the logarithmic functions in this table. The explicit calculations show, that their complexities are close to the maximal possible complexity of a binary function (for a given value of n). For a primitive residue a modulo p we dene the logarithm of the residue k by the Fermat formula aloga (k) = k (mod p) . Reducing this integer modulo 2, we construct the binary function with the values x(k) = loga (k) (mod 2) Z/(2Z)

(for the arguments values k = 1, 2, . . . , p 1).

Example. p = 7, a = 3 provide log3 k = (0, 2, 1, 4, 5, 3), x(k) = (0, 0, 1, 0, 1, 1). Thus we obtain for each a the sequence of the n = p 1 logarithms reduced modulo 2, x Zn . Now we shall apply the complexity, dened by 2 the graphs of Theorem 1, to this binary sequence x. Theorem 2. The modulo 2 reduced logarithms x of the consecutive residues modulo p have the following values (for p 13) in terms of the dierences graphs of the preceding theorem (for n = p 1)x=1

p=3x=6

(x T4 , a = 2);

p=5

(x T8 , a = 2 or 3);x = 11

p=7

(x T16 , a = 3 or 5);

p = 11

216 O30

285 360

(x = 285 (O30 T16), a = 2, 6, 7 or 8);

40


3546

p = 13

1238 2857 3450 1935

1206 1647 2488 2737 4050

O12

O12

(x = 1238 (O12 T16), a = 2);

(x = 1206 (O12 T16), a = 11).

In this description we denote the binary sequence x = (x1 , . . . , xn ) Zn 2 by the binary decimal integer 2n1 x1 + 2n2 x2 + + xn (thus, x = 285 in the case p = 11, n = 10 means the sequence (0, 1, 0, 0, 0, 1, 1, 1, 0, 1) of the 10 binary digits of the number 285 = 256 + 16 + 8 + 4 + 1). The proofs of the statements of Theorem 2 are nite, but long calculations. In the simplest case p = 3, a = 2 the geometrical progression {2k } = 1, 2 (mod 3) for k = 1, 2 implies the logarithms log2 1 = 0 , log2 2 = 1 .

The reduced logarithms sequence is (x1 = 0, x2 = 1), providing the position of x in the graph of the theorem, according to formula (4.1). For higher values of p (especially for p = 11 and 13) the calculations are longer. To accelerate them it is useful to reduce the length n cyclical sequences x modulo the n cyclic rotations group Z/(nZ) (identifying, say, the sequences (0100) and (0001) for n = 4). For this reduction by the action of it is useful to consider x as a residue modulo 2n 1. Since the operator A commutes with these rotations, this reduction accelerates the calculations (about n times). It is also useful to calculate rst the kernel of At (say, for large t). This kernel is represented by the vertices, forming a binary tree. To obtain the attracted trees of the cycles it is sucient to add to the points of the cycle this subspace of Zn . This reasoning explains the homo2 geneity of the graphs of the table of Theorem 1: all the attracted trees are

Chapter 4

41

isomorphic to the above kernel (the union of the cycles being the image of the linear operator At for large t). These long calculations lead to the table of Theorem 2, whose comparison with the table of Theorem 1 shows, that the complexity of the logarithmic function almost attains the maximal value, possible for any binary function on a set of n points. This observation follows from both theorems only in the case n 12, but it might be considered for larger values of n, at least as a probable conjecture. Similar theorems (and conjectures) should be also considered for the nonbinary functions, for instance, for the functions, whose values are residues modulo some integer q: x (Z/qZ)n , xk Z q .

Unfortunately, I cant guess the answers in such a generalization of Theorem 1 (even for q = 2, as above): the table of Theorem 1 shows a rather irregular dependence on n, and even the averaged asymptotics for n are interesting, but unknown. All this set of theorems and conjectures can be extended to the general Galois elds almost literally, but I had restricted myself above to the simplest case of the residues eld Zp (and sometimes only to the binary base Z2 ), since the missing complexity theory should rst study this elementary case. The author is grateful to Prof. Shparlinski (Departement of Computing, Macquarie University, Sydney), who, having read the original draft of the present book, had proved some of the conjectures, discussed above, correcting also some misprints in my previous publications of some others.

42


Chapter 5 Adiabatic study of distribution of geometrical progressions of residuesI shall describe here some physical arguments, explaining the asymptotical equipartition of the sequence of the residues of the division by p of the members of the geometrical progression {Ak , 1 k p} among all the nonzero residues of the division by p, A being a primitive residue and the number , 0 < < 1, being xed, while p tends to innity. We shall try to evaluate the number N of those residues of the progressions members, which belong to the interval G : 1 c p

of the values of the residues c. To make it, we shall use the logarithms, transforming the geometrical progression into an arithmetical one: ln Ak =k

= ka

(where a = ln A) .

The condition Ak (mod p) G can be written in terms of the logarithms as the inclusion of the number k into one of the intervals of the following system (represented in gure 5.1):k

43

j ,

44

Adiabatic study of equipartition.

where j is the interval between the logarithms of members of two arithmetical progressions, j = { : ln(jp) < ln(jp + p)} . The length of the interval j equals |j | = ln (for large j).k,min

jp + p = ln jp

j+ j

= ln 1 +

j

j

ka

(k + 1)a

k,max

ln(pj)

ln(p(j + 1))

ln(p(j + ))

Figure 5.1: Adiabatic approximation of a nonuniform sequence of the logarithms of the members of an arithmetical progression. The sum D() of the lengths of all these intervals is a quantity of order (1/j), where the numbers j of the intervals, forming this nite sum, is xed by the maximal and minimal logarithms k of the members of our geometrical progression. This leads to an approximate relation D() ln(jmax ) (where jmax for p ) ,

the approximation word meaning the smallness of the relative error. The total length of the whole interval of the axis , containing all our logarithms k , is D( = 1) ln(jmax ). The arithmetical progression { k } is uniformly distributed along the axis (according to the H. Weyl theorem on the equipartition of the fractional parts of an arithmetical progression in the interval (0, 1)). This leads to the guess, that the number N of the visits of the points k to the union of the intervals j should be asymptotically proportional to the

j

j+1

Chapter 5

45

fraction, formed by the lengths sum of these intervals in the length of the whole segment of the axis , that is it should be asymptotically proportional to the ratio D()/D(1). We arrive, therefore, to the conclusion, that for p one should expect the asymptotically behaviour of the visits number N p D() D(1) ,

which means the asymptotical equipartition of the sequence of the residues of the division of the members {Ak , 1 k p} by p (the domain G, the visit to which we counted, forming the -th part of Zp ). Remark 1. The next term of the development ln(1 + x) x x + . . . (for 2 small x) leads to the prediction, that one should expect the upper decline from the asymptotical value, N > (p 1).2

Remark 2. Our reasoning (far from the mathematical rigour) might be considered as some kind of adiabatic replacement of the logarithmically nonuniform sequence (of the numbers ln(jp) = ln p + ln j and of the intervals j , starting at these points) by the arithmetical progressions (of numbers and of intervals correspondingly) see gure 5.1 at page 44. In fact the step ln(j + 1) ln j = ln j+1 1/j of the logarithmical j sequence is slightly declining when j is growing, and therefore this logarithmical sequence is not exactly an arithmetical progression (being, however, close to it for rather long intervals of change of j, provided that the step of the approximating arithmetical progression had been chosen correspondingly to the range of j). Were the logarithmical sequence an arithmetical progression, our reasoning would be based strictly by the Weyl fractional parts equidistribution theorem. Thus, the remaining foundation of our euristical reasoning depends on the evaluation of the error of the adiabatic approximation (which might also be replaced by the modication of the Weyls theorem proof, including the study of the behavior of the Fourier coecients of the characteristical function of the union of the intervals j ). For p = 997, = 1/2, = 1/2, A = 7, I had calculated the visits number N to be 279, which is larger than the conjectured asymptotical expression, (p 1) 249. For p = 1009, A = 11 (and = = 1/2) the visits number is N = 269 (for the visits of the residues 11k , 1 k 503, modulo 1009 to the domain

46


G = {1 x 504} (mod 1009)). The asymptotical expression provides (p 1) 252 (suggesting the decline of the dierence between N/m and |G|/z like c/ p). It would be interesting to see more examples, for larger p, to evaluate the decline empirically. The quantities N (A)/m, corresponding to dierent primitive elements A, may deviate from the mean value for some exceptional values of A, and the evaluation of the dispersion of this deviation might provide an interesting information on the rarity of those exceptional values of A for large primes p. A dierent approach to the asymptotical equipartition of the sequence of the residues for the division by p of the members of the geometrical progression {Ak , 1 k p} among all the residues of the division by p (A being a primitive residue, the number being xed, 0 < < 1, and the prime number p tending to innity) is provided by the following construction. Consider the multiplicative group of complex number Zp = {z C : z p = 1} . The functions on this group are the (Fourier) linear combinations of the characters, e0 1, e1 = z , e2 = z 2 , . . . , ep1 = z p1 .

The multiplication by A of the residues (of the division by p) can be represented in these notations as the mapping f : Zp Zp , where f (x) = xA . This mapping acts on the functions as the linear operator (and algebra morphism) f ek = eAk . Function e0 is invariant under this mapping, while the remaining p 1 characters are permuted cyclically (A being a primitive element). To prove the equipartition one have to prove the convergence to zero of the time averages of the functions, orthogonal to e0 . For the character ek we have to study the time averageT 1

ek =t=0

(f )t ek /T .

To study this averaged function, make once more the Fourier transform, ordering the harmonics ek (k = 0) in the order of their numbers in the above

Chapter 5

47

cyclic permutation of order p 1 (that is in the order of the sequence of the residues At (mod (p 1)) in Zp1 ). To do it we consider the multiplicative group of complex numbers, Zp1 = {wt = e2it/(p1) } , where 0 t < p 1. The corresponding harmonics E0 , . . . , Ep2 : Zp1 C are dened by the formula Er (w) = w r . As it is explained above, we identify the characters ek (k = 0) with these functions Er in the way that the sequence e1 , f e1 , (f )2 e1 , . . . , (f )p2 e1 takes the form E0 , E1 , E2 , . . . , Ep2 , and therefore f takes the form Er Er+1 of the multiplication by the function w(= E1 ). For the time average we obtain the expression Er = 1 wT 1 1 (1 + w + w 2 + + w T 1 )Er = Er , T T w1

tending to zero for large T (for every value of r and hence for all the characters ek , k = 0). Thus, the time average of any function on the group Zp1 tends to its space average for T . The space average of the harmonic ek along Zp1 is easy to compute: 1 p1p1

(f ) ekt=0

t

1 = p1

p1

k=1

ek =

e0 , p1

since p1 ek = 0 (by the Vieta theorem for the equation z p = 1). k=0 Therefore, the mean value ek tends to 0 for p if k = 0. Thus the time average along the segment of geometrical progression {At } of any xed linear combination of the harmonics on group Zp tends for p to its space average. Applying this to the characteristic function of part G of group Zp , we would prove the asymptotical equipartition statement for the progression segment at the limit p .

48


This reasoning is, however, unsucient for a rigorous proof, since the characteristic function of domain G is not a xed linear combination of the harmonics: the number of the harmonical summands, needed to approximate this characteristic function of G on Zp is growing with p. Remark. In our study of the asymptotical equipartition for p we had xed the (Jordan measurable) domain G in the real continuous torus T n , evaluating the number N of visits of a sequence {Ak } to the corresponding domain G(p) of the nite n-torus Zn ; this domain consists of a nite number p of points (growing with p). Perhaps, together with the natural domains G (similar to the band 0 u d in T 2 ), one might take also many more complicated domains G(p) (similar, say, to the set dened by the condition that u is even for the points (u, v) Zn ). p The conjecture is that the equipartition asymptotics would hold for p even for such irregular domains, provided that the algorithm, dening the domain G(p), would be suciently simply. I do not know any proved theorem of this kind (which would be interesting even for the distribution of the fractional parts of an arithmetical progression members on a circle), while one might consider as some conrmation of this conjecture the Skolem theorem on the zeroes of the recurrent sequences {ak }: this theorem claims, that the set {t : at = 0} consists of a nite set of arithmetical progressions of integers t, whatever be the recurrent sequence. The ergodic theory content of the conjectured chaoticity theorems is the statement, that a sucient chaoticity of the dynamics obstructs the prediction of all those properties of the trajectories, which might be computed by simple algorithms. Remark. The study of the equipartition of the nite segments of geometrical progressions along the one-dimensional nite torus Zp irregular domains, described above, might be useful for the investigation of the degree of the equipartition of the geometrical progressions segments along nite elds with a large number z = pn of elements, living on an n-dimensional torus. The point is, that the mapping k Ak is sending bijectively the set of the nonzero residues (of the division by the number z 1) on the set of the nonzero elements of the z elements eld (living on an n-torus). Therefore, knowing the equipartition property for the arithmetical progression {tr, t = 1, 2, 3, . . . } on the 1-dimensional nite torus Zz1 one might deduce the information on the partition of the geometrical progression {B t , t =

Chapter 5

49

1, 2, 3, . . . }, where B = Ar , along the nite n-torus Zn (consisting of z = pn p points). For instance, to study the asymptotical behaviour of the number N of the visits of a segment of geometrical progression {B t } to domain G of the eld of z elements, it would suce to know the asymptotical behaviour of the number of the visits of the corresponding segment of arithmetical progression {tr, t = 1, 2, 3, . . . } to the full preimage f 1 G of domain G under the above bijection k Ak (which we have denoted here by f ). In this sense to prove the asymptotical equipartition of the segments {B t } of geometrical progressions along the n-dimensional (nite) torus it would suce to prove the proportionality of the number of visits to subdomains of the nite circle Zz1 by the arithmetic progression {tr} segment to the measure of the subdomain, but one needs to know this approximate proportionality inequality for (dierent from the ordinary intervals) subdomains of the nite circle, provided that the subdomain is dened by an algorithm of bounded complexity (like, say, the domain f 1 (G) for the domain G, that we are studying on the nite torus). While the arithmetical progressions are a much simpler object for equipartition study, than the geometrical progressions, the known results on the arithmetical progressions are unsucient for two reasons: one needs to study the visits to complicated domains f 1 (G) and the one-dimensional nite torus Zz1 points number z 1 = pn 1 is not prime. In spite of this formal inadequacy of the equipartition results on the nite circles to our problem, it seems that it might be not too dicult to attain the relevant arithmetical progressions equipartition results that are needed for the geometrical progressions equipartition studies along the nite n-tori.

50


Chapter 6 Projective structures, generated by a Galois eldThe Galois elds algebra has a remarkable geometrical aspect, similar to the projective geometry aspect of linear algebra (including the ellipsoids and hyperboloids principal axis geometry instead of the quadratic forms eigenvalues theory). Calculations are usually simpler in the algebraic version, but the real understanding is reached only by the geometrical approach to the principal axis theory. Goethe told, that mathematicians are similar to French people: they translate everything in their language, and it changes all the content. I shall describe now the geometrical version of the Galois eld algebra: the theory of the projective structures on nite sets and the study of the action on them of the groups of the Frobenius transformations (of powers calculations) k (x) = xk . Recall rst the projective line notation. Consider all the straight lines, containing point 0, on the usual plane R2 . The manifold of all such lines is one-dimensional, and it is dieomorphic to the circle. To see it, describe the points of the plane by their Cartesian coordinates (u, v). The line is then described by its equation, u = v, where is a constant (depending on the chosen line and dening it) see gure 6.1. The constant is called the ane coordinate on the projective line (whose 51

52u

Projective study of Galois table.u = v

0

1

v

Figure 6.1: A projective line and its ane coordinate . points are the straight lines of the plane, going through the origin 0). However, like the whole Earth sphere is not represented on one hemisphere map, the ane coordinate values do not provide all the straight lines, going through the origin. Namely, they do not provide the vertical line (v = 0 at gure 6.1). Therefore one adds an innite point = to the axis of the variable , providing the description of the whole real projective line RP 1 by the values RP 1 { R, = } .

This innite point = of the projective line (representing the vertical line v = 0 of gure 6.1) is as good as all the others, since the vertical line on the plane is as good as the others (for instance, they are transformed to any of them by the plane rotations). A dierent choice of the initial coordinates system on the plane (say, one might take u = v, v = u) would produce a dierent ane coordinate = u/ in our example) and a dierent point ( = ) on the projective ( v line RP 1 (which would correspond in our example to the horizontal line, v = 0, of gure 6.1). In the neighbourhood of the vertical line of gure 6.1, (where = 0 in our example) the new ane coordinate ( = 1/ in our example) is regularly parametrizing the real projective line. Therefore, the tending of the ane coordinate to + or leads to the same place ( = 0) of the manifold of the straight lines of the plane, containing the origin. This compact manifold, the real projective line, is therefore dieomorphic to the circle (gure 6.2): RP 1 S 1 .

Chapter 6 =0

53

+

=0

Figure 6.2: The real projective line dieomorphism to a circle. For a dierent linear coordinates choice on the plane (which might be nonorthogonal) the new ane coordinate would be a fractionally-linear function of the old one, a + b , (6.1) = c + d since the new coordinates have the form u = au + bv , v = cu + dv .

The new coordinate axis should not coincide, therefore ad = bc. The transformation of the axis of , dened by the formula (6.1) is called a projective transformation. The axis of is considered here as being completed by the innite point, and the formula (6.1) denes a dieomorphism of the real projective line (that is of a circle) to itself. Of course, the algebraic versions of this simple geometry are the conventions = for c + d = 0 , = a/c for = . The real projective space of dimension n 1, RP n1 = (Rn 0)/(R 0) ,

is dened similarly to the projective line, being the manifold of the straight lines, going through the origin 0 of the n-dimensional vector space Rn . Its ane chart is constructed from a linear coordinates system (u1 , . . . , un ) in Rn : if un = 0, we dene the vector Rn1 , whose coordinates are 1 = u1 /un , . . . , n1 = un1 /un .

(it is important to observe the denominators independence on the number j of coordinate j ). The geometrical meaning of these algebraic formulae is that they describe the projective transformations which are obtained (say, for m = 2) when one projects one plane P in the 3-space onto another plane P by the rays, starting at a common projection center (gure 6.3).

54

In other terms, we take the intersection point of the line that we wish to describe with the hyperplane un = 1 of space Rn , this point is the image of the line on the ane chart Rn1 (similar to the situation of gure 6.1 at page 52, where n = 2). To obtain all the straight lines of space Rn , going through the origin, one has to use n such ane charts (represented by the n hyperplanes, {un = 0}, . . . , {u1 = 0}. Thus, for n = 2 one needs two charts ( and at gure 6.2 of page 53). The corresponding fractionally-linear transformations are described in RP m by the following extension of formula (6.1): for j = 1, . . . , m the coordinates of the image point of the point with ane coordinates (1 , . . . , m ) are aj,1 1 + + aj,m m j = b 1 1 + + b m m

Therefore, the theory that we are describing is basic both for the geometry"" ! "! "! "! "! "! "! "! "! "! "! "! "! "! "! "! "! "! "! "! "! ""!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"! !""""""""""""""""""""! "!!!!!!!!!!!!!!!!!!!!!" ""!" !!""!""!""!""!""!""!""!""!""!""!""!""!""!""!""!""!""!""!""!""! !!!!!!!!!!!!!!!!!!!! !"""""""""""""""""""""!"! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! "!"""""""""""""""""""""" !!!!!!!!!!!!!!!!!!!!! """"""""""""""""""""! "!!!!!!!!!!!!!!!!!!!!!" "!" !!!!!!!!!!!!!!!!!!!!!! !"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!" """"""""""""""""""""""! "!!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!""!" !!!!!!!!!!!!!!!!!!!!!! """"""""""""""""""""! ""!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!""! """"""""""""""""""""""!" !!!!!!!!!!!!!!!!!!!!!! !"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!" !"""""""""""""""""""""! "!!!!!!!!!!!!!!!!!!!!"!" !"!!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"!"! """""""""""""""""""""" !!!!!!!!!!!!!!!!!!!!! !"!0

Figure 6.3: A projective transformation of a cat.P

Projective study of Galois table.

P

Chapter 6

55

of the projections, sending the straight lines to the straight lines, and to the theory of the perspective (where the rails of a straight railway meet at an innitely far point on the horizon). The great Italian painter Paolo Uccello (whose name means Bird), being one of the rst painters, seriously studying the mathematical theory of the correct perspective drawing, answered to his wife, inviting him at midnight to the bedroom I am coming what a nice perspective, having in mind the beauty of his remarkable drawing. To become familiar with this projective geometry, one might try to prove the following facts: 1. The real projective plane is unorientable, the real projective spaces RP m being orientable for odd dimensions m and nonorientable for the even dimensions. 2. The complement to a small disc on the real projective plane is diffeomorphic to the Mbius band (which surface had been discovered by o Mbius, due to this fact). o The complex projective spaces CP m are dened similarly to the real one, starting from the complex vector space Cn (where n is still equal to m + 1). The points of this complex projective space are the complex straight lines, going through the origin of space Cn : CP n1 = (Cn 0)/(C 0) .

The ane coordinates and the projective transformations are dened by the same formulae, as in the real case1 . In the complex projective spaces and transformations cases the diculty, described above, vanishes. Thus, the complex projective line, CP 1 , is obtainable from the axis of the complex variable by the addition of one innite point. The resulting variety is dieomorphic to the ordinary sphere S 2 and is called the Riemann sphere. In the neighbourhood of the innite point = the ane complex coordinate is the function = 1/.This is the important advantage of algebra: the algebraists are ready to apply their formulae to the objects, which are quite dierent from those, for which these formulae had been proved, and if the result occurs to be wrong, they postulate it to be presumably true for their ideal objects, replacing the dicult reality study by the easier investigations of these ideal objects.1

56


The complex straight lines, going through the origin 0 of the complex space Cn , intersect the sphere, centered at the origin (which is dened by the equation |uk |2 = 1 and is dieomorphic to the sphere S 2n1 of the real vector space R2n ) along the real circles S 1 . The real lines in Rn , going through zero intersect the sphere S m (where m = n 1, |uk |2 = 1) along 0-dimensional spheres S 0 (each of which consists of two opposite points). While the real projective space RP m might be obtained from the sphere S m by the gluings of all pairs of opposite points, which are proportional vectors of the real vector space, RP m = S m /(S 0 = {1}) , the complex projective space manifold, CP m might be obtained from the sphere S 2n1 (where n = m + 1), gluing at one point each circle S 1 , along which the sphere intersects a complex straight line, going through the origin. All the points of this circle are complex proportional vectors of the complex vector space, and they are obtainable from (arbitrary) one of the points of this circle, multiplying it by all the complex number of modulus one: CP m = S 2m+1 /(S 1 = {ei }) . The really 4-dimensional manifold CP 2 is obtainable from the ane complex plane C2 by the addition at the innity of a complex projective line CP 1 , that is of a Riemann sphere: CP 2 = C2 S 2 . The complex projective line CP 1 might also be described as the manifold, whose points are the special great circle S 1 S 3 . A special great circle is the intersection of the 3-sphere S 3 with a complex straight line in C2 , containing the origin 0. This circle is the union of the points of sphere S 3 which are complex proportional: CP 1 = S 3 /(S 1 = {ei }) . It is interesting to notice, that dierent special great circles in the 3-dimensional sphere S 3 , which is bred into these circles, are situated in the sphere specially: the linking coecient of any pair of such special great circles is equal to one.

Chapter 6

57

The linking coecient of two disjoint oriented smooth closed curves in the oriented 3-sphere (or in the oriented Euclidean 3-space) is dened as the intersection index of one of these circles with the smooth oriented immersed compact surface, whose boundary is the other circle (gure 6.4).

Figure 6.4: Two curves, having the linking coecient equal to two. The orientations are used here the following way: the orienting frame of the surface at a boundary point consists of the orienting vector of the boundary curve, followed by the tangent vector to the surface, directed inside it. The intersection index of the second curve with the surface, whose bound is the rst curve, is counting the intersection points of these two objects, equipped with the signs, which are positive, if the 3-frame, formed by the three vectors, orienting the curve and the surface, orients positively the 3space. The linking coecient of two curves does not depend on the choice of the surface, bounded by one of the curves: the surface should only be nowhere tangent to the second curve. Linking coecient L of two oriented curves is symmetric with respect to the curves ordering: L(I, II) = L(II, I). The 3-sphere bration, whose bers are the above special great circles, is called Hopf bration S 3 (CP 1 = S 2 ), whose ber is S 1 . This bration is the basic object for many branches of mathematics. The 3-sphere is the ordinary Euclidean 3-space, compactied by the addition of one point. In this model of the 3-sphere S 3 , the Hopf bration becomes the decomposition of the Euclidean 3-space into a straight line (originated from the circle, containing the added point) and the complement to this line, bered into closed curves, whose pairwise linking coecients are all equal to

58


one. While I am able to draw the resulting nice picture, I shall not show it in this lecture, leaving to the listener the pleasure to draw it themselves. Instead we shall transfer now the theory, which we had described above in the real and in the complex case, to the case where the numbers are replaced by the residues of the division by a prime number p: we shall dene the nite projective spaces P m (Zp ), very similar to the real manifolds RP m and to the complex manifolds CP m (but consisting of nite numbers of points). We shall start from the nite projective line, P 1 (Zp ). It is dened as the set of the straight lines of nite plane Z2 , containing the origin: p P 1 (Zp ) = (Z2 p 0)/(Zp 0) .

In terms of the coordinates (u, v) on the nite plane (which are now residues modulo p) the straight line equation has the form u = v, but the ane coordinate Zp is also a residue modulo p. To attain all the straight lines, one should add to these p values of the ane coordinate one more value, denoted by the symbol (to include the vertical line v = 0, taking into account that = u/v for v dierent from 0). Therefore, the nite projective line P 1 (Zp ) consists of p + 1 points: |P 1 (Zp )| = p + 1 ( = 1, 2, . . . , p; ) . The projective transformations a + b , c + d (6.2)

(where a, b, c and d belong to Zp , ad bc being dierent from zero) permute some way the p+1 points of the nite projective line, but these permutations are not arbitrary. Indeed, the symmetric group S(p + 1) of all the permutations of p + 1 points of our nite projective line consists of (p + 1)! permutations, and the order of the group of the projective transformations (6.2) is much smaller. Lemma. The group PL(Zp ) of the projective transformations ( 6.2) of the nite projective line, consisting of p + 1 points, is formed by p(p2 1) permutations.

Chapter 6

59

Indeed, if a = 0, we might divide all the coecients by a, writing the transformation (6.2) in a form, where a = 1 The other coecients, and b c, may have (independently of each other) p values each, and the remaining b coecient, d, should verify the condition d = c, having (for xed values of and of c) p 1 possible values. This way we get p2 (p 1) transformations b (6.2). In the remaining case a = 0 the nondegeneracy condition would be bc = 0. If d = 0, we might, dividing by d, reduce the formula (6.2) to the form, where d = 1; the number of such transformations is (p 1)2 , since c = 0. b Finally, in the case a = d = 0, the transformation has the form b, such transformations number being p 1 (since b = 0). Thus, the total number of the projective transformations of the nite projective line P 1 (Zp ) of all the three kinds is equal to the sum p2 (p 1) + (p 1)2 + (p 1) = (p 1)(p2 + (p 1) + 1) = p(p2 1) , proving the Lemma. The factorial, (p + 1)!, is much larger, if p is large (starting already at p = 5, where (p + 1)! = 720, p(p2 1) = 120). The point is, that the subgroup of the projective permutations of the points of the nite projective line forms in the whole permutation group S(p + 1) a small subset of those permutations which preserve some remarkable geometrical structure in the nite set P 1 (Zp ), consisting of p + 1 points. Unfortunately, I have no geometrical description of this remarkable nite projective line structure, while algebraically the structure is described by an ane coordinate Zp on the complement to some innitely far situated point = of the nite set M , equipped with the projective structure. A dierent

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V. I. Arnold Dynamics, Statistics and Projective Geometry of Galois Fields