Transversal Intersection and Sum of Polynomial Ideals

TL;DR

This paper establishes conditions for the transversal intersection of polynomial ideals and applies these results to compute Betti numbers for specific ideal sums involving matrices and minors.

Contribution

It introduces new conditions for transversal intersection of polynomial ideals and applies them to compute Betti numbers for particular matrix-related ideals.

Findings

Derived conditions for transversal intersection of polynomial ideals

Computed Betti numbers for ideals involving matrices and minors

Provided examples illustrating the theoretical results

Abstract

In this paper we derive some conditions for transversal intersection of polynomial ideals. We exhibit some examples. Finally, as an application of the results proved, we compute the Betti numbers for ideals of the form $I_{1}(XY) + J$, where $X$ and $Y$ are matrices and $J$ is the ideal generated by the $2\times 2$ minors of the matrix consisting of any two rows of $X$.