Small Resultant Systems via Linear Combinations

2026-07-31

AI Summary

This work addresses the inefficiency of traditional methods for constructing resultant systems of polynomial systems, which typically yield an excessively large number of polynomials. The authors propose a novel approach that leverages linear combinations of the input polynomials to construct resultants, drastically reducing the system size. In the homogeneous case, they prove that the existence of a nontrivial common solution can be characterized using only ${d+n-1 \choose n-1}s - n^2 + 1$ polynomials. Moreover, when the number of variables is fixed, they provide an explicit construction of polynomial size. By integrating linear combinations, resultant theory, and coefficient matrix analysis from algebraic geometry, this method substantially improves the known upper bounds on resultant system size—even outperforming existing results in the bivariate case.

Abstract

For a system of $s$ homogeneous polynomials of degree $d$ in $n$ variables, say ${\bf{f}} = 0$, we consider the problem of constructing resultant systems. A resultant system is a finite set of polynomials in the coefficients of the input polynomials, the vanishing of which characterizes the systems $\bf{f}$ with a common non-zero solution. The classical approaches for constructing resultant systems rely either on maximal minors of large coefficient matrices or on the coefficients of a resultant of generic linear combinations of the input polynomials. Typically, they produce resultant systems containing a very large number of polynomials. We develop new constructions based on taking resultants of linear combinations of the input polynomials; this results in resultant systems of small cardinality. Our main results are: 1) We prove that a resultant system with ${d+n-1 \choose n-1} s-n^2+1$ polynomials exists; each polynomial is the resultant of $n$ linear combinations of the input polynomials. This improves the previously known upper bounds, even for systems of bivariate homogeneous polynomials. 2) Under the assumption that the input polynomials are non-zero, we construct explicit resultant systems with cardinality $\mathrm{poly}(s,d)$, when $n$ is fixed.

Problem

• resultant system
• homogeneous polynomials
• common non-zero solution
• linear combinations
• polynomial system