2026/07/31 by M. Levent Doğan, Elias Tsigaridas, Zafeirakis Zafeirakopoulos
Mathematics · Computer Science · #math.AC #cs.SC #math.AG
arxiv created 2026/07/31 · arxiv updated 2026/08/03
For a system of s homogeneous polynomials of degree d in n variables, say \bff = 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 \bff 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-n2+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 poly(s,d), when n is fixed.