In mathematics, the resultant of two monic polynomials and over a field is defined as the product
of the differences of their roots, where and take on values in the algebraic closure of . For non-monic polynomials with leading coefficients and , respectively, the above product is multiplied by
- and this expression remains unchanged if is reduced modulo . Note that, when non-monic, this includes the factor but still needs the factor .
- Let . The above idea can be continued by swapping the roles of and . However, has a set of roots different from that of . This can be resolved by writing as a determinant again, where has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient of appears.
- Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.
- If and , then
- If have the same degree and ,
- The resultant of a polynomial and its derivative is related to the discriminant.
- Resultants can be used in algebraic geometry to determine intersections. For example, let
- define algebraic curves in . If and are viewed as polynomials in with coefficients in , then the resultant of and gives a polynomial in whose roots are the -coordinates of the intersection of the curves.