Efficiently Evaluating U-Resultants

The objective is to efficiently evaluate u-resultants for numerical uvalues (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. ∗Supported by the NSF grant CCF 0430741