Bounds for degrees of syzygies of polynomials defining a grade two ideal

TL;DR

This paper establishes explicit exponential bounds on the degrees of polynomials in syzygy modules for grade two ideals, improving previous bounds and providing methods to compute bases with bounded degrees.

Contribution

It explicitly bounds the degrees of syzygies for grade two ideals and demonstrates how to compute a basis with these bounds, enhancing prior results.

Findings

Explicit exponential bounds on syzygy degrees for grade two ideals.

Method to compute a basis of the syzygy module with bounded degrees.

Improved bounds over previous results in known cases.

Abstract

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.