A subdivision-based algorithm for the sparse resultant

Abstract

Multivariate resultants generalize the Sylvester resultant o f t wo polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra.We propose a determinantal formula for the sparse resultant of an arbitrary system of n + 1 polynomials in n variables. This resultant generalizes the classical one and has signi cantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their B zout number. Our algorithm uses a mixed polyhedral subdivision of the Minkowski sum of the Newton polytopes in order to construct a Newton matrix. Its determinant is a nonzero multiple of the sparse resultant and the latter equals the GCD of at most n + 1 such determinants. This construction implies a restricted version of an e ective sparse Nullstellensatz.For an arbitrary specialization of the coe cients there are two methods which use one extra variable and yield the sparse resultant. This is the rst algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n. We conjecture its extension to producing an exact rational expression for the sparse resultant.

Sparse Rational Univariate Representation

“…For sparse resultant matrix constructions, E stands for the number of lattice points of the polytope that is the Minkowski sum of the Newton polytopes of the input polynomials, e.g. [12]. We can bound E using the volume of the Newton polytopes.…”

“…(10), (11), (12), and (19) we bound the degree and the height of the polynomials in the SRUR of the roots of (Σ) using the mixed volume. THEOREM 2.2.…”

Improved algorithms for computing determinants and resultants

“…Matrix construction is independent of the coefficient values and can thus be conducted off-line; its complexity can be asymptotically smaller than that of manipulating the matrix for system solving [CE00]. M is quasi-Toeplitz, i.e., its entries depend only on a À b; where a; b belong to two subsets of Z n which index, respectively, the rows and columns of M: The most relevant property of such matrices is that, by Emiris which extends Macaulay's classical result to the toric case, with S being a submatrix of the Sylvester-type resultant matrix M: This formula gives a general means for computing R exactly, and leads to output-sensitive bounds.…”

Toric border basis

“…In the context of elimination and resultant theory, the approach of Macaulay [14] for the construction of projective resultant matrices has been extended successfully to toric resultant for Laurent polynomials [9,1,8,21,2,5]. By analyzing the support of the Laurent polynomials, resultant matrices of smaller size than for the projective resultant can be constructed.…”