Upper bounds for regularity of radicals of ideals and arithmetic degrees

TL;DR

This paper establishes new upper bounds for the regularity of radicals of ideals and their arithmetic degrees in polynomial rings, advancing understanding of algebraic complexity related to ideal radicals.

Contribution

It provides explicit upper bounds for the regularity of radical ideals and their arithmetic degrees based on initial ideal properties and degrees.

Findings

Regularity of radical ideals is bounded by $d^{(n-1)2^{r-1}}$.

Arithmetic degree of ideals is bounded by $2d^{2^{n-r-1}}$.

Bounds are derived using properties of strongly stable and Borel type ideals.

Abstract

Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa an upper bound for the regularity of $\sqrt{I}$. More specifically we show that $\text{reg}(\sqrt{I})\leq d^{(n-1)2^{r-1}}$. In the second part, we show that the $r$-th arithmetic degree of $I$ is bounded above by $2\cdot d^{2^{n-r-1}}$. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.