Comptage des racines r

P(X) = Xn+a1Xn−1+· · · +an∈R[X]α1, . . . , αnPC
s0, s1, s2, . . . P s0=n
B=
s0s1. . . sn−1
s1s2. . . sn
sn−1sn. . . s2n−2
S(u) =
n−1
X
i,j=0
si+juiuj, u = (u0, . . . , un−1).
(s, t)S
P s−t s+t
B=tV V V α1,· · · , αn
S
S(u0, . . . , un−1)=(u0+u1α1+· · · +un−1αn−1
1)2+· · · + (u0+u1αn+· · · +un−1αn−1
n)2
P q l1, . . . , lq
S=Pq
k=1 mkl2
kmk
q
S q
P s −t
lk¯ lk
v w lk=v+iw ¯ lk=v−iw
mk(l2
k+¯
l2
k) = 2mkv22mkw2.
B P =X2+bX +c P =X3+pX +q

B P
det(B)=∆P=Y
i<j
(αi−αj)2.
σk
σk(α1, . . . , αn) = X
1≤i1<i2<···<ik≤n
αi1αi2· · · αik.
σk= (−1)kak
k≥n sk−σ1sk−1+σ2sk−2+· · · + (−1)nσnsk−n= 0
k≤n sk−σ1sk−1+σ2sk−2+· · · + (−1)kkσk= 0
T Q(T) = TnP(1/T )
Q(T) =
n
X
j=0
(−1)jσjTj=
n
Y
i=1
(1 −T αi),
σ0= 1
T
T Q0(T) =
n
X
j=0
(−1)jjσjTj=−(
n
X
k=1
T αk
1−T αk
)(
n
Y
i=1
(1 −T αi))
= (
∞
X
k=1
skTk)(
n
X
i=0
(−1)i−1σiTi)=(
∞
X
k=1
skTk)(−Q(T))
q(x, y, z) = x2−2y2−z2−4yz + 2xz
q(x, y, z) = xy +yz +zx