Sparse Resultant of Composed Polynomials II Unmixed–Mixed Case

This paper is the second in a series of papers on sparse resultants of composed polynomials. In the first paper, “Sparse Resultant of Composed Polynomials I", Hong and Minimair (2000, http://minimair.org/HM2000.ps) considered the sparse resultant of polynomials having arbitrary (mixed) supports composed with polynomials having the same (unmixed) supports. Here, we consider the sparse resultant of polynomials having the same (unmixed) supports composed with polynomials having arbitrary (mixed) supports (under a mild technical assumption on their exponents). The main contribution of this paper is to show that the sparse resultant of these composed polynomials is the product of certain powers of the (sparse) resultants of the component polynomials. The resulting formula looks similar to the formula of the first paper, which is good because it suggests that there is some common underlying structure for sparse resultants of composed polynomials. However, the formulae differ substantially in detail. It also seems that it is not possible to apply the techniques used to show the main result of the first paper in order to show the formula of the present paper. It is expected that this result can be applied to compute sparse resultants of composed polynomials with improved efficiency.