Moral (but false) result
If gcd(A, B) = 1 then r  ]{[ρ]: ∆(ρ) = 0 et A(ρ) ∈ (Q(ρ)∗
)2
}.
False: if A = k2
∈ Q∗
then for nρ = ordρ(B2
− 4k2
C) we have
X
[ρ]
nρP[ρ] =
X
ρ
nρP(ρ) = 0
indeed
P
ρ nρ[P(ρ)] − d[O] is a divisor of Y − kT − B(X)
2k .
Notations:
• ∆∗
is the radical of ∆ (so ∆∗
is squarefree) ;
• Ω(∆) is the number of irr. factors of ∆ with multiplicities ;
• M∆,A = resY (∆(Y ), X2
− A(Y )) ;
• (·) is the characteristic non-zeros squares where · naturally lives.
Proposition
If gcd(A, B) = 1 then
]{[ρ]: ∆∗
(ρ) = 0 et (A(ρ)) = 1} = Ω(M∆∗,A) − Ω(∆∗
) − 2
If P(A) divides ∆∗
for a P of degree 2.