Rhine Workshop on Computer Algebra
Proceedings of the RWCA 2006 (Basel, Switzerland)
University of Basel, 2006, 181-187
Abstract
The objective is to efficiently evaluate u-resultants
for numerical u-values (such as over a finite field).
The
u-resultant of n homogeneous polynomials in n+1 variables is
defined to be the multi-variable resultant
of these n polynomials and a general linear form in the same variables
whose coefficients are represented by the symbols u0,...,un.
It is shown that the u-resultant can be
extracted from a matrix that is smaller than the
standard Macaulay matrix obtained from the definition of the u-resultant.
The ratio of the sizes of the
standard Macaulay matrix and of the matrix introduced by the current paper
approximately equals the average of the total degrees
of the homogeneous polynomials. As expected, experimental timings show
a substantial speed-up when using the smaller matrix.