This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the @math -resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the monomial structure of any given polynomial system. We thus obtain a fast new algorithm for univariate reduction and a better understanding of the underlying projections. As a corollary, we show that a refinement of Hilbert's Tenth Problem is decidable within single-exponential time. We also show how certain multisymmetric functions of the roots of polynomial systems can be calculated with sparse resultants.
From an applied angle, our observations on degeneracies and handling polynomial systems with infinitely many roots nicely complement the work of Emiris and Canny @cite_43 . In particular, their sparse resultant based algorithms for polynomial system solving can now be made to work even when problem B occurs. Also, an added benefit of working torically (as opposed to the classical approach of working in projective space) is the increased efficiency of the sparse resultant: the resulting matrix calculations (for polynomial system solving) are much smaller and faster. In particular, whereas it was remarked in @cite_41 that Gr "obner basis methods are likely to be faster than the GCP for sparse polynomial systems, the toric GCP appears to be far more competitive in such a comparison.
@cite_41 @cite_43
1976392590 2066130115
Multipolynomial resultants provide the most efficient methods known (in terms as asymptoticcomplexity) for solving certain systems of polynomial equations or eliminating variables (, 1988). The resultant of f"1, ..., f"n in K[x"1,...,x"m] will be a polynomial in m-n+1 variables which is zero when the system f"1=0 has a solution in ^m ( the algebraic closure of K). Thus the resultant defines a projection operator from ^m to ^(^m^-^n^+^1^). However, resultants are only exact conditions for homogeneous systems, and in the affine case just mentioned, the resultant may be zero even if the system has no affine solution. This is most serious when the solution set of the system of polynomials has ''excess components'' (components of dimension >m-n), which may not even be affine, since these cause the resultant to vanish identically. In this paper we describe a projection operator which is not identically zero, but which is guaranteed to vanish on all the proper (dimension=m-n) components of the system f"i=0. Thus it fills the role of a general affine projection operator or variable elimination ''black box'' which can be used for arbitrary polynomial systems. The construction is based on a generalisation of the characteristic polynomial of a linear system to polynomial systems. As a corollary, we give a single-exponential time method for finding all the isolated solution points of a system of polynomials, even in the presence of infinitely many solutions, at infinity or elsewhere. Abstract We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. The matrix determinant is a non-trivial multiple of the sparse resultant from which the sparse resultant itself can be recovered. The algorithm is incremental in the sense that successively larger matrices are constructed until one is found with the above properties. For multigraded systems, the new algorithm produces optimal matrices, i.e. expresses the sparse resultant as a single determinant. An implementation of the algorithm is described and experimental results are presented. In addition, we propose an efficient algorithm for computing the mixed volume of n polynomials in n variables. This computation provides an upper bound on the number of common isolated roots. A publicly available implementation of the algorithm is presented and empirical results are reported which suggest that it is the fastest mixed volume code to date.