114 - Anneau des séries formelles. Applications - IMJ-PRG

A
A[[X]] + .×
A[X]→A[[X]] A
Xn= (0,...,1,0, . . . )
v∀x, y ∈A[[X]]
v(x) = +∞ ⇐⇒ x= 0
v(x) = v(−x)
v(x+y)≥min(v(x), v(y))
A∀a∈A∗, v(a.x) = v(x)
A[[X]] A[X]
Xn
A v(xy) = v(x) + v(y)A[[X]]
Pn≥0anXna0∈A∗
1
1−X=Pn≥0Xn
A=⇒A[[X]]

D(X)=1 v0(D(x)) ≥v0(x)−1D(x) =
0 =⇒x∈k k
{i|ai,n 6= 0}nPSiSi→0
S=PanXnT v0(T)≥1
(anTn)nS◦T=PanTn
S T
k[[X]] (Xi) (X)
exp
log exp
(1 + X)α= exp(αlog(1 + X))
(1 + X)α
(un)nu0, . . . , uk−1
un=a1un−1+· · · +akun−kG(X)
Pn≥0unXnP(X) 1 −Pk
i=1 aiXiP(X).G(X) =
−Pn<k Pi≤nun−iaiXnP(X)
G(X) = 1
1−X−X2
Fn=1
√5(ϕn−ϕ0n)

{1, . . . , n}n
α1n1+· · · +αpnp=n
K
F(a, b, c, X) = Pn≥0hainhbin
hcinn!Xn
hain=(a+n−1)!
(a−1)! X(1−X)F00+c−(a+b−1)XF0−abF = 0
C∞
C∞(I, R)→R[[X]]
f7→ X
n≥0
f(n)(0)
n!Xn