In UnivariatePolynomialCategory:
if R has CommutativeRing then
discriminant : % -> R
resultant : (%, %) -> R
However, in SparseUnivariatePolynomial:
if R has IntegralDomain then
discriminant(p) == discriminant(p)$PseudoRemainderSequence(R, %)
resultant(p1, p2) == resultant(p1, p2)$PseudoRemainderSequence(R, %)
So discriminant and resultant are not defined for domains that
are COMRING but not INTDOM, for example IntegerMod 6.
The resultant can be solved easily:
In BezoutMatrix, I recently added subResultant which
computes resultant by determinant of Sylvester matrix.
subResultant(p, q, 0) = resultant(p, q)
So we can add function resultant in BezoutMatrix, and
in SUP, redirect to BezoutMatrix when coefficients are
COMRING but not INTDOM.
As for discriminant, wikipedia says it can be computed
by take determinant of Sylvester matrix with first column
replaced by 1:
https://en.wikipedia.org/wiki/Discriminant#Discriminant_of_a_polynomial_over_a_commutative_ring
My concerns are performance, the determinant method
is very slow compared with pseudo division, but I think
there's no better algorithm for COMRING? If so, I'll start
implementing such functions in bezout.spad and poly.spad.
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