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.
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.
If A ∈ (Q∗
)2
then r = Ω(∆∗
) − 1.
We have r ≤ deg(∆∗
) ≤ 5 and all the rank are possible.