# 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](../../references.md#cite-gcl), 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 $P$).