SI Maths2010 English

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Selection Internationale 2010

Scientific test : Mathematics

Exercise 1 : A few results on polynomial interpolation

Let f : [a, b] → R be a continuous function. Let n ∈ N, and n + 1 points x0 < x1 < · · · < xn in[a, b], all distinct, in increasing order.

1) Show that there is one and only one polynomial pn with degree less than n satisfying

pn(xi) = f (xi), i = 0, . . . , n .

2) We assume from now on that f is n + 1 times differentiable over [a, b]. Show that for anyx ∈ [a, b], there is a point ξ x ∈ ] min(x, xi), max(x, xi)[ such that

f (x) − pn(x) =1

(n + 1)!πn+1(x) f (n+1)(ξ x),

where πn+1(t) :=n

i=0(t − xi). Deduce from this identity the upper bound

sup[a,b]

|f − pn| ≤ (b − a)n+1

(n + 1)!sup[a,b]

|f (n+1)|.

3) Let R > b−a2 , and c := a+b

2 . We assume (only for this question) that f is defined over ]c−R, c+R[,and given there by a power series centered at c:

f (x) =+∞k=0

ak(x − c)k, for all x ∈ ]c − R, c + R[.

Show that for R large enough, the sequence ( pn)n∈N converges uniformly to f over [a, b].

4) We assume (only for this question) that the points xi are uniformly distributed:

xi := a + ih, h :=b − a

n

.

a) Show that: sup[a,b] |πn+1| ≤ hn+1 maxs∈[0,n] |s(s − 1) · · · (s − n)|b) Using Stirling’s formula: n! ∼ √

2πnne

n, deduce from this inequality: there exists C > 0

(independent of n) such that

sup[a,b]

|πn+1| ≤ C

b − a

e

n+1

Deduce from this inequlaity an upper bound on sup[a,b] |f − pn|, better than the one given in 2).

c) Show that there exists c > 0 (independent of n) such that: sup[a,b] |πn+1| ≥ cn

b−ae

n+1.

5) We define, for n ∈ N, t ∈ R, T n(t) := cos(n arccos t). Show that T n is a polynomial with degreen, satisfying the induction relation

T n+1(t) = 2t T n(t) − T n−1(t), n ∈ N, t ∈ R.

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Show that T n has n roots y0 < y1 < · · · < yn in [−1, 1], all real and distinct. Determine these roots.

6) We assume (only for this question) that xi := a+b2 + b−a

2 yi. Show that sup[a,b] |πn+1| =

2b−a4

n+1, and deduce from this identity an upper bound for sup[a,b] |f − pn|, better than the one

given in 2).

7) Let Rn[X ] the set of all polynomials with real coefficients and degree less than n. Show that

there is p ∈ Rn[X ] such that

sup[a,b]

|f − p| = inf q∈Rn[X ]

sup[a,b]

|f − q |

Such a polynomial p will be called a polynomial of best approximation of f .

8) We say that g ∈ C ([a, b]) equioscillates on (k + 1) points of [a, b] if there are points x0 < x1 <

· · · < xk in [a, b] such that

For all i = 0, . . . , k , |g(xi)| = g and for all i = 0, . . . , k − 1, g(xi+1) = −g(xi).

Show that if p is a polynomial of best approximation , it equioscillates on n + 2 points in [a, b].

9) Deduce from the previous question that there is only one polynomial of best approximation.

10) Show that there is a unique unitary polynomial p with degree n + 1 which has minimal uniformnorm over [−1, 1] ( p = sup[−1,1] .| p|). Determine this polynomial.

Exercise 2

Let K be a field of characteristic different from 2. Recall that a quadratic form (of finite dimension)over K is given by a finite dimensional K-vector space V and a map q : V

→K such that

i) q (λv) = λ2v for all λ ∈ K and v ∈ V ;

ii) the map q : V × V → K defined by (x, y) → q (x, y) =1

2

q (x + y) − q (x) − q (y)

is bilinear.

Also recall that q → q is a bijection from the set of quadratic forms over K onto the symetricbilinear forms. In the following, all quadratic forms will be non-degenerate , i.e., for all v ∈ V −{0},there exists w ∈ V such that q (v, w) = 0. An isometry between two quadratic forms (V, q ), (V , q )

is a bijective linear map f : V → V such that q ◦ f = q .

For all a1, . . . , an ∈ K − {0}, we denote a1, . . . , an the quadratic form defined over Kn by the

formula q ((x1, . . . , xn)) = a1x21 + · · · + anx

2n. Finally we denote Q(K) the set of non-degenerate

quadratic forms of finite dimension.

Isotropic forms, isometries and orthogonal sum.

We write q ∼= q if q and q are isometric.

1) Show that ∼= is an equivalence relation on Q(K).

2) A quadratic form q is said to be isotropic if there exists x = 0 such that q (x) = 0. Otherwise, q is said to be anisotropic .

a) Prove that for q

∈ Q(K), there exists x

∈V such that q (x)

= 0.

b) Denote H the quadratic form over K2 defined by H((x1, x2)) = x1x2. Show that any isotropicquadratic form q ∈ Q(K) of dimension 2 is isometric to H and deduce that, for all λ ∈ K − {0},one has H ∼= λ, −λ.

2



3) A basis (e1, . . . , en) of V is said to be orthogonal for q if q (ei, e j) = 0 for all i = j. Prove thatfor any quadratic form q ∈ Q(K), there exists an orthogonal basis (one can use question 2)a) andconsider {x}⊥ = {y ∈ V / q (x, y) = 0}). Deduce that there exists a1, . . . , an ∈ K − {0} such thatq ∼= a1, . . . , an.

4) Assume q (x) = 0. Prove that the map y → sx(y) = y − 2 q(x,y)q(x) x is an isometry. Deduce that if

q (v) = q (w) = 0, then there exists an isometry u from q onto itself such that u(v) = w.

5) Let (V, q ), (V , q ) be two quadratic forms in Q(K). Define a quadratic form q ⊥ q : V × V → K

by the formula q ⊥ q ((x, x)) = q (x) + q (x) for all x ∈ V, x ∈ V .

a) Show that q ⊥ q is in Q(K).

b) Show that if q 1, q 2, q 3 satisfy q 1 ⊥ q 3 ∼= q 2 ⊥ q 3, then q 1 ∼= q 2 (one can argue by induction anduse question 4)).

II: Pfister forms and similarity factors.

Denote im(q ) = {a ∈ K− {0} / ∃x ∈ V q (x) = a} and Sim(q ) = {a ∈ K− {0} / aq ∼= q }.

1) Prove that Sim(q ) is a subgroup of K− {0} that contains all squares of K− {0}.

2) Prove that for all quadratic forms q , there exist a unique integer m and an anisotropic quadraticform q an, unique up to an isometry, such that q ∼= q an ⊥ mH where mH = H ⊥ · · · ⊥ H. (One canuse Part I).

3) Show that if q is isotropic, im(q ) = K− {0}.

For all a1, . . . , an ∈ K−{0} and all quadratic forms (V, q ), we denote a1, . . . , an⊗ q the quadraticform a1q ⊥ · · · ⊥ anq over V n. A Pfister form is a quadratic form of the form

a1, . . . , an := 1, −a1 ⊗ · · · ⊗ 1, −an.

4) Show that for all a,b,λ ∈ K−{0}, one has a, b ∼= −ab,a + b and a, b ∼= a, (λ2−a)b.

5) Let p = a1, . . . , an be a Pfister form.

a) Show that 1 ∈ im( p) and deduce that there exists φ unique up to an isometry such that p = 1 ⊥ −φ.

b) Let b1 ∈ im(φ). Prove that there exists b2, . . . , bn ∈ K − {0} such that p = a1, . . . , an ∼=b1, . . . , bn.

c) Let x ∈ V be such that p(x) = 0. Show that p ∼= p(x) p and deduce that Sim( p) = im( p).

III: Application to the level of a field.

The level of a field K is the least integer n(K) such that −1 is a sum of n(K) squares in K. If suchan integer does not exist, we set n(K) = +∞.

1) Compute the level of the field Q of rational numbers, of the field C of complex numbers, and of the field Z/pZ for p an odd prime number.

2) Prove that if n(K) is finite, then n(K) is an (integer) power of 2. (Hint: one can introduce anadequate Pfister form and use Part II).

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