The height of the mixed sparse resultant

On the height of the Sylvester Resultant

Let n be a positive integer. We consider the Sylvester Resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is a generic polynomial of degree n. If f is a quadratic polynomial, we find the resultant's height. If f is a cubic polynomial, we find tight asymptotics for the resultant's height.

Koszul-type determinantal formulas for families of mixed multilinear systems

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kin…

The Canny-Emiris conjecture for the sparse resultant

We present a product formula for the initial parts of the sparse resultant associated to an arbitrary family of supports, generalising a previous result by Sturmfels. This allows to compute the homogeneities and degrees of the sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain a similar product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated to a mixed subdivision of a polytope. Applying these results, we prove that the sparse resultant can be computed as the quotient of the determ…

Generic fiber rings of mixed power series/polynomial rings

Let K be a field, m and n positive integers, and X = {x_1,...,x_n}, and Y = {y_1,..., y_m} sets of independent variables over K. Let A be the polynomial ring K[X] localized at (X). We prove that every prime ideal P in A^ = K[[X]] that is maximal with respect to P\cap A = (0) has height n-1. We consider the mixed power series/polynomial rings B := K[[X]][Y]_{(X,Y)} and C := K[Y]_{(Y)}[[X]]. For each prime ideal P of B^ = C that is maximal with respect to either P \cap B = (0) or P \cap C = (0), we prove that P has height n+m-2. We also prove that each prime ideal P of K[[X, Y]] that is maximal …

基于稀疏结式的一般求解器

A General Solver Based on Sparse Resultants

本文利用稀疏结式构造，提出一种不增加问题维度的新方法将求根问题转化为特征值问题，并实现了一个通用多项式系统求解器，在视觉、机器人和结构生物学应用中展示了高效率和准确性。

Mixed multiplicities and projective degrees of rational maps

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.