AGRÉGATIONCONCOURS EXTERNE ’E’E’E

AGRÉGATION

CONCOURS EXTERNE

E ’ ’ E ’ E

m i n i s t è r e é d u c a t i o n n a t i o n a l e j e u n e s s e vie associative

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EAE MAT 1

SESSION 2012 ____

AGRÉGATION CONCOURS EXTERNE

Section : MATHÉMATIQUES

COMPOSITION DE MATHÉMATIQUES GÉNÉRALES

RECTIFICATIF

Dans la Partie IV, on prendra K = C.

2 R^d

N Z Q

R C

x x X Card(X)

X Y

X Y ={x∈X |x∈Y }.

X R^d λ

λX ={y∈R^d| ∃x∈X, y =λx}.

f(n) =P(n) n ∈N

P₀, . . . , P_N−1 ∈C[T] n∈N f(n) =P_r

N(n)(n)

r_N(n) n N

d m₁, . . . , m_d

n∈N

u_n= Card({(k₁, . . . , k_d)∈N^d|

d

i=1

m_ik_i =n}) v_n=_n

i=0u_i

Le symbole



(v_n)_n∈N d= 1 n→v_n

i∈ {1, . . . , d} U_i =

k∈NT^kmi

U =

n∈N

u_nTⁿ∈Z[[T]], u_n

U U_i

U ×_d

i=1(1−T^mi).

V =

n∈Nv_nTⁿ

V U

n1na_nTⁿ⁻¹ (F₁F₂) = F₁F₂ +F₁F₂ F₁

F₂ F F⁽⁰⁾ = F

F^(k+1) F^(k)

G=

n∈NTⁿ

G² G G

k ∈N G^k+1 G^(k)

(u_n)_n∈N (v_n)_n∈N

m₁ =· · · = m_d = 1 n → v_n

v : N → Z n → v_n

1 2

p q x = p/q

(u_n)_n∈N

u_n= Card(Z∩[0, nx])−nx n ∈N

a, b, c, d∈Z A= (a, b)∈Z² B = (c, d)∈Z² [A, B]

R² A B

Card([A, B]∩Z²) = pgcd(c−a, d−b) + 1.

R² X R

4

X^o P R² P

Z²

Card(P∩Z²) =V(P) + 1

2Card(∂P ∩Z²) + 1.

a, b, c, d∈Z a < b c < d [a, b]×[c, d]

a, b (0,0) (a,0) (0, b)

R²

P P

P

P₁ P₂ S₁ S₂

Z² P₁ P₂ P₁∩P₂

[A, B] A B S₁∩S₂

[A, B] =∂P₁∩∂P₂ P₁

(AB)

P₁ P₂ P₁∪P₂

P₁ P₁∪P₂ P₂

A B C R²

{A, B, C}

P S

P N S

A ∈S A

S {A}

A, B, C, D S (α, β, γ)

D (A, B, C) α+β+γ = 1

α, β, γ

N 3 i→A_i {1, . . . , N}

S A₁ S

(A₂A₃)

N 4 M P

{A₂, . . . , A_N} M {A₁, A₂, A₃}

M A₁, . . . , A_i

i

6

K d

K[X₁, . . . , X_d] _d

i=1X_i^ai d a =

(a₁, . . . , a_d)∈N^d M N^∗

m= (m₁, . . . , m_d)∈N^∗d P ∈K[X₁, . . . , X_d]

π_m(P) = deg(P(T^m1, . . . , T^md))

deg(0) = −∞ n ∈ N H_m,n

P ∈C[X₁, . . . , X_d]

P(T^m1X₁, . . . , T^mdX_d) =TⁿP(X₁, . . . , X_d) K[X₁, . . . , X_d][T]

P K[X₁, . . . , X_d] n ∈ N P

H_m,n

H_m,n K

N N n

H_m,n

K[X₁, . . . , X_d] H_m,n n

N

N C g₀ G

V d C π G V

π : G → GL(V)

(e₁, . . . , e_d) V α₁, . . . , α_d π(g₀)(e_i) = ξ^αie_i i∈ {1, . . . , d}

P ∈ C[X₁, . . . , X_d] x = (x₁, . . . , x_d) ∈ C^d P(x) =

P(x₁, . . . , x_d) u C C^d

u C C[X₁, . . . , X_d]

u(P)(u(x)) = P(x)

P ∈C[X₁, . . . , X_d] x∈C^d P u(P )

π(g₀)(e_i) =ξ^mie_i τ =π(g₀) C[X₁, . . . , X_d] A={P ∈C[X₁, . . . , X_d]|τ(P) =P }.

A C[X₁, . . . , X_d]

A