Diophantine Sets Over Polynomial Rings. by Martin Davis; Hilary PutnamReview by: H. B. EndertonThe Journal of Symbolic Logic, Vol. 37, No. 3 (Sep., 1972), pp. 602-603Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272756 .

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602 REVIEWS

(though this abstract by Myhill is partly erroneous in a way that Putnam explains) a number of decision problems connected with Hilbert's tenth problem are shown to be unsolvable. There is no algorithm by which, when a polynomial P with integer coefficients is given, we can always determine whether:

(1) (3X)(Yj)x(Y2)x ... (Yk)x * P(x, Y1 , Y2, Yk) # 0;

(2) P represents every natural number; (3) P represents every sufficiently large natural number; (4) P represents every integer.

In (1) the bound variables range over the natural numbers; and a subscript after a quantifier indicates an upper bound, so that e.g. " (Yl)x " is to be read as "for every yj less than or equal to x." ALONZO CHURCH

MARTIN DAVIS. Applications of recursive function theory to number theory. Recursive function theory, Proceedings of symposia in pure mathematics, vol. 5, American Mathematical Society, Providence 1962, pp. 135-138.

This paper is largely expository. Let q be the class of Diophantine sets, J be the class of exponential Diophantine sets, and R be the class of recursively enumerable sets. Clearly, 9_ a _ aM. It was shown in Davis, Putnam, and Robinson, XXXV 151, that J = M. Earlier, XX 182 showed that -9 = b' if the following hypothesis is true.

H. There exists a binary Diophantine relation S such that S(x, y) -m y < E(x) for some exponential function E but there is no polynomial bound on y in terms of x for S(x, y). (Here exponential function is any function which can be expressed explicitly in terms of +, *, and exponentiation.)

Thus, if H is true, there is no decision method for determining the solvability of Diophantine equations. The author with Putnam has shown that certain propositions imply H. If H is false each of the propositions must also be false which would yield new results. In this paper, three such propositions are stated. We give the first two of these, correcting the second as the author has done in the paper reviewed next below.

1. For each k, there are x, y, m such that x3 my3 = 1 and x > mk. 2. Let an , a' be successive solutions of the Pell equation x2 - dy2 = 1. There exists a value

of d such that a' cannot be written in the form alaj(r2 + ds2) unless n is a power of 2. JULIA ROBINSON

MARTIN DAVIS. One equation to rule them all. Transactions of the New York Academy Of Sciences, ser. 2 vol. 30 no. 6 (1968), pp. 766-773.

The author gives yet another assertion of number theory whose truth, while an open question, would imply the recursive unsolvability of Hilbert's tenth problem. (Cf. the preceding review.) The assertion is that the equation 9(U2 + 7v2)2 - 7(r2 + 7s2)2 = 2 has no solution in the natural numbers except for the trivial solution u = r = 1, v = s = 0. H. B. ENDERTON

MARTIN DAVIS. Extensions and corollaries of recent work on Hilbert's tenth problem. Illinois journal of mathematics, vol. 7 (1963), pp. 246-250.

This paper consists of three separate notes, each related in some way to the 1961 paper of Davis, Putnam, and Robinson (XXXV 151). The first note gives a small class . of functions such that every partial recursive function f is of the form f(x) = U(G.y(S(e, x, y) = T(e, x, y))) for functions U, S, and Tin .2. .2 is the closure under composition of the set containing 2X, x y, 0,

Ii, , K, and L, where I?(xl, - * *, x,,) = xi and K and L are pairing functions. The second note observes that in 1949 Quine raised the question of the decidability of the set

of universal sentences true in the natural numbers with addition, multiplication, and exponen- tiation (XVI 76). The results of XXXV 151 immediately imply that (as Quine conjectured) this set is undecidable.

The third note concerns the representation of recursively enumerable sets in "almost- Diophantine" form. H. B. ENDERTON

MARTIN DAvIs and HILARY PUTNAM. Diophantine sets over polynomial rings. Ibid., pp. 251-256.

This paper establishes the recursive unsolvability of a problem analogous to Hilbert's tenth

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REVIEWS 603

problem. The main result is that every recursively enumerable set of positive integers is Diophan- tine over the ring ZRX] of formal polynomials with integer coefficients. (A set is Diophantine over a ring R if it has the form {x:(3y1 , e * , yX e R)P(x, Yi, ,** *, y n) = O} for a polynomial P over R.) The proof utilizes results of Julia Robinson's 1952 paper XX 182.

H. B. ENDERTON

JULIA ROBINSON. Diophantine decision problems. Studies in number theory, edited by W. J. LeVeque, Studies in mathematics, vol. 6, The Mathematical Association of America, Washing- ton, D.C., distributed by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1969, pp. 76-116.

This expository paper presents a self-contained account of the work on the decision problem for Diophantine equations (Hilbert's tenth problem) up to 1969. The presentation is efficiently organized and nicely written. It does not presuppose a background in logic or recursive function theory on the part of the reader.

The author first discusses the concept of an effective procedure in an informal and intuitive manner, speaking of "listable" sets and "computable" functions and sets. This discussion is followed by a formal definition of recursive enumerability and recursiveness, and a summary of the reasons for accepting Church's thesis.

The first specialized result is a version (due to Davis; see XXIII 432, page 113) of the fact that recursive sets are arithmetically definable: All (and only) recursively enumerable relations are of the form {x: (3y)(Vu : y)(3v1 c y) ... (3Vk < y)P = 0} where P is a polynomial over the integers. This theorem is then employed in a proof of the main result of XXXV 151, that all recursively enumerable relations are exponential Diophantine.

Finally, results from the author's XX 182 are proved; it is shown that if any relation of expo- nential order of growth is Diophantine, then all recursively enumerable relations are Diophantine. The question whether any such Diophantine relation exists was, of course, still open when this paper was published. H. B. ENDERTON

JULiA ROBINSON. Unsolvable diophantine problems. Proceedings of the American Mathe- matical Society, vol. 22 (1969), PP. 534-538.

The results established in this paper neatly extend those of the well-known paper of Davis, Putnam, and the author, The decision problem for exponential diophantine equations (XXXV 151), and of more recent work of Davis and the author (see the preceding reviews). In what follows, all variables are over the natural numbers.

A relation R(x1, * * *, x,,) is Diophantine in a set S iff there is a polynomial P with integer coefficients such that R(x1 , * * * , Xn) (3y, , * *, Yk)(3z1, , * *, zX)(P(xi ,* * *, Xn, Y1 I e * X ,A Z1,***, zj) = O & z1 e S &... & z, S). A function f is Diophantine if the graph of f is a Diophantine set. The author proves: (Theorem 1) If M is an infinite set and H is a Diophan- tine function such that for all m e M, (m, H(m)) = 1 and 2H(^) = 1 (mod m), then r - 2t is Diophantine in M; (Theorem 2) x = 22" for n > 0 iff x is a power of 2 and there are a, v such that x = 1 + 3(u2 + v2); (Theorem 3) for every r.e. (recursively enumerable) set S, there is a polynomial P with integer coefficients such that S = {x I (3RY , X - *, y* )(3t)P(x, yj , * * *, Yn) = 2%}. A corollary to Theorem 1 is that every r.e. set is Diophantine in the set of numbers of the form 22". From Theorem 3 it follows that there is no algorithm to decide of an arbitrary poly- nomial P(x ,*** *, x,) whether it assumes a power of 2 as a value, and that every r.e. set is Diophantine in the set of powers of 2. In the proof of these results once again the theory of the Pell equation X2 - (a2 _ 1)y2 = 1 and the Chinese remainder theorem are employed.

The larger context of this paper is the attempt to prove that Hilbert's tenth problem is recur- sively unsolvable by showing that all r.e. sets are Diophantine. This attempt began with the paper XVIII 341 of Davis. In the main, the approach to the problem has been a search for a Diophantine relation R(x, y) of "exponential order of growth"; that is, a Diophantine relation R(x, y) satisfying (1) (Vx, y)(R(x, y) -- y S xX) and (2) (Vk)(3x, y)(R(x, y) & y > xk). This approach was introduced by the author in her basic XX 182. Subsequently Davis and Putnam showed that if there is such a Diophantine relation and if for each n > 0 there are n primes in arithmetic progression, then every r.e. set is Diophantine (Davis and Putnam, A computational proofprocedure; axioms for number theory; research on Hilbert's tenth problem, Rensselaer Poly-

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