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TSI2
R[X]Rn[X]
n n
Rn[X]
fR[X]
∀P∈R[X], f(P) = (2X−1)P0+ (X2−X−2)P00.
fR[X]
n∈NfRn[X]fnRn[X]
MnfnRn[X]
fnfn
fnfnfn
n>2λ0, λ1, . . . , λnfn
λ06λ16. . . 6λn
λkk k ∈[[0, n]]
P fnλk
k
P
(Pk)06k6nRn[X]
k∈[[0, n]] Pkk f
n= 3
M3f3R3[X]f3
P0, P1, P2P3
Pk
Rn[X]
ϕ
∀(P, Q)∈R[X]2, ϕ(P, Q) = Z2
−1
P(t)Q(t)t.
P t 7→ P(t)
ϕR[X]
ϕ(P, Q) = hP, QiP Q R[X]
∀(P, Q)∈R[X]2,hf(P), Qi=−Z2
−1
(t2−t−2)P0(t)Q0(t)t.

∀(P, Q)∈R[X]2,hf(P), Qi=hP, f (Q)i.
n∈N∗(Pk)06k6n
h., .i
k∈N∗Pk∈Rk−1[X]⊥
X2R1[X]
R1[X] = Vect(1, X)
n2ERn
•Id E
f g E f ◦g f g
f0= Id f1=f k 2
fk=f◦f◦. . . ◦f
| {z }
k
•f E
f a E a, f(a), f2(a), . . . , fn−1(a)
E
n= 2 f a E
a, f(a)E
f a E
a, f(a), f2(a)E
•PR[X] deg(P)P
Rn−1[X]
E=R2
αR2R2
A=4−6
1−1
a= (2,3) α(a)α
α2(a)x y
α2(a) = xa +yα(a)
A0αa, α(a)
2α
bR2b, α(b)R2
bR2

TSI2
E=R3βR3
R3B=
2 0 0
0 4 −6
0 1 −1
β
βR3
β
βR3
β D =
2 0 0
0 2 0
0 0 1
β2−3β+ 2 Id = 0 0 R3
β
E=Rn−1[X]n
2
γRn−1[X]PRn−1[X]
P0
γRn−1[X]
γX2−3X+ 1= 2X−3
γXn−1γkXn−1k
1n−1
k
γ
n2
δRn−1[X]PRn−1[X]
Q Q(X) = P(X+ 1) −P(X)
δRn−1[X]
δX2−3X+ 1=(X+ 1)2−3(X+ 1) + 1−X2−3X+ 1
Q0, Q1, . . . , Qn−1Rn−1[X]
i0n−1 deg(Qi) = i
Q0, Q1, . . . , Qn−1Rn−1[X]
δ
k1n−1
δXk
k−1
PRn−1[X]P
1
deg δ(P)= deg(P)−1
δ
Xn−1, δXn−1, δ2Xn−1,· · · , δn−1Xn−1.
δ
δ

δRn−2[X]
δRn−2[X]
P0, P1, P2, . . . , Pn−1
δ
P0, P1, P2, . . . , Pn−1Rn−1[X]
P0(X) = 1 , P1(X) = 1
1!X , P2(X) = 1 2!X(X−1), . . .
Pn−1(X) = 1
(n−1)!X(X−1)(X−2) · · · (X−n+ 2).
j1n−1
Pj(X) = 1
j!X(X−1)(X−2) · · · (X−j+ 1) = 1 j!
j−1
Y
k=0
(X−k).
P0, P1, P2, . . . , Pn−1Rn−1[X]
n= 4
X3−5X2+X−3P0, P1, P2, P3
R3[X]
0P1, P2P31
P2P3
i j i 6= 0 1 6j6n−1
δPj=Pj−1δiPji6j i > j
δ