Sparse resultant of composed polynomials II: unmixed-mixed case
J. Symbolic Comput. 33 (2002), no. 4, 467-478
Abstract
This paper is the second one in a series of papers on
sparse resultants of composed polynomials.
In the first paper, ``Sparse Resultant of Composed Polynomials I'',
the author and Hoon Hong 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 details.
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.