ccp2012 banqoral

θ∈Rn∈N∗C[X]
R[X]
P=X2n−2Xncos (nθ)+1.
P= 3X4−9X3+ 7X2−3X+ 2 Q=X4−3X3+ 3X2−3X+ 2
P Q R[X]C[X]
P Q
P Q
C[X]P= 2X4−3X2+ 1 Q=X3+ 3X2+ 3X+ 2
PC[X]P
QC[X]Q
U V P U +
QV = 1
U V
R=X5+X4
(X−2)2(X+ 1)2.
R
x7−→ R(x) ] −1; 2[
EK=R C
n f E f (P) = P−P0
f
f
f
Q∈E. P f (P) = Q
P∈E P (n+1)
A=1 2
2 4 fM2(R)f(M) = AM.
f
f
f f.

A B n AB BA
A B k ∈NAkBk
Mn(C)n
A= (ai,j )16i6n
16j6n∈ Mn(C)kAk= sup
16i6n
16j6n
|ai,j |
kABk6nkAkkBkp>1kApk6np−1kAkp
A∈ Mn(C)XAp
p!
ΦRn[X]P(X)Φ 7−→ P(X)−P(X−1)
ΦRn[X] Φ Φ
ER C f g E f ◦g=
(g◦f) = f
(g◦f) = g
E=f⊕g
n>1. n
A=
2−1 0 ··· 0
−1 2 −1
0−1 0
2−1
0··· 0−1 2
n>1DnA
Dn+2 = 2Dn+1 −Dn
Dnn
A0A

E n R(ei)E v1, v2, . . . , vnn
E
f E
∀i∈ {1; 2; . . . ;n}, f(ei) = vi.
L(E)EMn(R)
n×n u L(E)ϕ(u) = (ei)u
(ei)u u (ei)
ϕL(E)Mn(R)
L(E)
E n RL(E)
EMn(R)n×nL(E)
+◦ Mn(R) + ×
◦ L(E)×
Mn(R)
(ei)E u L(E)ϕ(u) = (ei)u(ei)u
u(ei)
ϕL(E)Mn(R)
u∈ L(E),(ei)(u◦u◦ ··· ◦ u
| {z }
n
) = (ei)un
E n
{e1, e2, . . . , en}E i = 2,3, . . . , n {e1+ei, e2, . . . , en}
E
E
E
f E n
E=f⊕f=⇒f=f2
f=f2⇐⇒ f=f2
f=f2=⇒E=f⊕f
EKn f E
L(E)EL(E)
n, f, f2,··· , fn2of