Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula

Abstract

In the case of univariate polynomials, the Bezoutian ${{\left( {P\left( x \right) - P\left( y \right)} \right)} \over {\left( {x - y} \right)}}$ defines a quadratic form of maximal rank whose signature is 1 when the degree is odd and 0 when it is even (*). More generally the expression ${{\left( {Q\left( y \right)P\left( x \right) - Q\left( x \right)P\left( y \right)} \right)} \over {\left( {x - y} \right)}}$ defines a quadratic form whose signature is the Cauchy index of the rational function Q/P. The Kronecker symbol or global residue is the linear form l associating to f (reduced modulo P) its coefficient of degree d - 1 (where d is the degree of P). When the polynomial P has only simple roots the Kronecker symbol (or global residue) of f is the number $\sum {{{f\left( p \right)} \mathord{\left/ {\vphantom {{f\left( p \right)} {P'}}} \right.} {P'}}} \left( p \right)$ and the signature of the quadratic form l(Qf ²) is again the Cauchy index of Q/P.

Partially supported by POSSO, Esprit BRA 846

Deutsche Forschungsgemeinschaft

Supported by grant KBN 2/1125/91/01