Resultants and Discriminants

Resultant(f, g, i): RngMPolElt, RngMPolElt, RngIntElt -> RngMPolElt

Resultant(f, g, v): RngMPolElt, RngMPolElt, RngMPolElt -> RngMPolElt

The resultant of multivariate polynomials \(f\) and \(g\) in \(P=R[x_1,\ldots, x_n]\) with respect to the variable \(v=x_i\), which is by definition the determinant of the Sylvester matrix for \(f\) and \(g\) when considered as polynomials in the single variable \(x_i\). The result will be an element of \(P\) again. The coefficient ring \(R\) must be a domain. There are two ways to indicate with respect to which variable the integral is to be taken: either one specifies \(i\), the integer \(1\leq i\leq n\) that is the number of the variable (upon creation of \(P\), corresponding to P.i) or the variable \(v\) itself (as an element of \(P\)). The algorithm used is the modular interpolation method, as given in [Geddes et al., 1992, pp. 412--413].

Discriminant(f, i): RngMPolElt, RngIntElt -> RngMPolElt

Discriminant(f, v): RngMPolElt, RngMPolElt -> RngMPolElt

The discriminant \(D\) of \(f\in R[x_1, \ldots, x_n]\) is returned, where \(f\) is considered as a polynomial in \(v=x_i\). The result will be an element of \(P\) again. The coefficient ring \(R\) must be a domain. There are two ways to indicate with respect to which variable the integral is to be taken: either one specifies \(i\), the integer \(1\leq i\leq n\) that is the number of the variable (upon creation of \(P\), corresponding to P.i) or the variable \(v\) itself (as an element of \(P\)).