Polynomial systems admitting a simultaneous solution

TL;DR

The paper provides a complete ideal-theoretic description of the resultant locus for $n$ univariate polynomials of degree $d$, via cascading matrices $M_k$ and their minors. It proves that $I_{d,n}$ is generated by all $(d+k)\times(d+k)$ minors of $M_k$ for $1\le k\le d$, using a diagonal-leading-term Gröbner-basis argument to identify the initial ideal. For $n=2$ this reduces to the classical Sylvester resultant, and the work connects to set-theoretic results of Orsinger-Kakie and Jouanolou through a saturation showing the projective equivalence of the relevant determinantal ideals. In addition to generators, the paper computes the dimension and degree of the variety $X_{d,n}$ and places the construction in a Koszul-resolution framework, delivering explicit, computable equations for the simultaneous-root problem.

Abstract

We provide a complete description of the ideal that serves as the resultant ideal for n univariate polynomials of degree d. We in particular describe a set of generators of this resultant ideal arising as maximal minors of a set of cascading matrices formed from the coefficients of the polynomials, generalising the classical Sylvester resultant of two polynomials.

Theorems & Definitions (24)

• Proposition 1
• proof
• Corollary 2
• Remark 3
• Theorem 4
• Remark 5
• Proposition 6
• Remark 7
• Remark 8
• proof : Proof of Proposition \ref{['prop:ltdiag']}