DM 29 - Alain Camanes

Partie I : Définition et propriétés
p, n q P =
p
P
k=0
akXkQ=
q
P
k=0
bkXk
C[X]apbq6= 0 P Q Res(P, Q)
Res(P, Q) =
a0b0
a1b1
b0
apa0b1
a1bq
apbq
.
p+q q
P p Q
P= 1 + 2X+ 3X2Q= 4 + 5X+ 6X2+ 7X3
Res(P, Q) = 
10040
21054
32165
03276
00307
.
Res(P, Q)MP,Q
Res(P, Q) = detMP,Q.
E=Cq−1[X]×Cp−1[X]F=Cp+q−1[X]u E F
(A, B)∈E u(A, B) = P A +QB
1. Res(P, Q) Res(Q, P )
2. Cas où uest bijective.
a) u
b) u P Q
c) P Q Ker u u
3. Matrice de u.
B= ((1,0),(X, 0),...,(Xq−1,0),(0,1),(0, X),...,(0, Xp−1)) EB0=
(1, X, . . . , Xp+q−1)F
a) uB B0
b) Res(P, Q)6= 0 P Q

4. Racine multiple.
a) PC[X]C
Res(P, P 0)=0
b) a, b
(a, b)X3+aX +bC
Partie II : Applications
5. Équation de Bézout. P=X4+X3+ 1 Q=X3−X+ 1
a) P Q
b) (A0, B0)C[X]P A0+QB0= 1
u
c) (A, B)∈C[X]2P A +BQ = 1
6. Équation de courbe.
(O, −→ i , −→ j) Γ
M t ∈R(x(t), y(t)) M
(x(t) = t2
t2+1
y(t) = 3t
t4+1
a) A, B, C, D (x, y, t)∈R3
P(t) = B(t)x−A(t)Q(t) = D(t)y−C(t)M(x, y)
t
(x=A(t)
B(t)
y=C(t)
D(t)
,∀t∈R
P Q
b) Γ
7. Nombre algébrique. P(X) = X2−3Qy(X)=(y−X)2−7
4√3 + √7