Theorem
r = Ω(M∆∗,A) − Ω(∆∗
) − (A) − Υ∆,A
− (2 deg(gcd(A, B)∗
) − Ω(Mgcd(A,B)∗,C) + Υgcd(A,B),C)
− (2 deg(gcd(A, B, C)∗
) − Ω(gcd(A, B, C)∗
))
where ΥP,Q small, explicit and often 0.
ΥP1,P2
:=























δt(1 − δu + (u) − (uDP1
)) si deg(P1) = 2 and P2(X) = Q(X)P1(X) + (tX +
u) with t, u ∈ Q, 0 6= Q ∈ Q[X] ;
ΥU,W + 2 −
Ω(MU,W ) + Ω(U(x2
)) −
Ω(U(
x2−DP2
4s ))
si deg(P1) = 4, deg(P2) = 2 and
P1(X) = U(P2(X)) for aU ∈ Q[X] and where
W (X) := 4sX2
+ DP2
X ;
0 otherwise.
with DP the discriminant of P and s leading coefficient of P2.
Algorithmic and explicit result
Corollary
If A = 0 then
r = Ω(MB∗,C) − Ω(B∗
) − 2 deg(gcd(B, C)∗
) + Ω(gcd(B, C)∗
) − ΥB,C.