A chain rule for multivariable resultants

We present a chain rule for the multivariable resultant, which is similar to the familiar chain rule for the Jacobian matrix. Specifically, given two homogeneous polynomial maps Kn _ K'n for a commutative ring K, such that their composition is a homogeneous polynomial map, the resultant of the composition is the product of appropriate powers of resultants of the individual maps. The chain rule for Jacobians is well known. There are some similarities between Jacobians and resultants. In [ 1 1], [ 12] and [ 1 3], the chain rule for resultants is proven for two polynomials in one variable as well as two homogeneous polynomials in two variables. (In [7], the chain rule is also proven for two homogeneous polynomials in two variables.) In this paper, a chain rule for the multivariable resultant is presented which generalizes the chain rule for resultants to n variables. The result is "universal" because the polynomials have indeterminate coefficients. The proof is more "conceptual" since it is based on the four axioms characterizing the multivariable resultant. In Section 1 the main result is stated and proved. In Section 2 consequences of the main result are derived; these include generalizations of the previous work on the chain rule of resultants. In Section 3 an axiomatic characterization of the multivariable resultant is included for the convenience of the reader.