Products of Ideals and Jet Schemes

TL;DR

This paper provides a comprehensive description of jet schemes for polynomial ideals, specifically the product of variables, and applies this to questions about ideal products and intersections relevant to algebraic differential equations.

Contribution

It offers a complete characterization of jet schemes for the ideal (x_1...x_n) and connects this to recent algorithmic questions in algebraic differential equations.

Findings

Explicit description of jet schemes for the ideal (x_1...x_n)

Insights into products and intersections of ideals in algebraic differential equations

Applications to algorithmic problems in algebraic geometry

Abstract

In the present paper, we give a full description of the jet schemes of the polynomial ideal $\left( x_1\ldots x_n \right) \in k[x_1, \ldots, x_n]$ over a field of zero characteristic. We use this description to answer questions about products and intersections of ideals emerged recently in algorithmic studies of algebraic differential equations.