# On the Betti numbers and Rees algebras of ideals with linear powers

## TL;DR

This paper investigates the Betti numbers and Rees algebras of ideals with linear powers, providing explicit polynomial formulas for certain classes and exploring the quadratic generation of Rees ideals related to matroid conjectures.

## Contribution

It explicitly computes Betti numbers for specific ideals with linear powers and examines the generators of their Rees ideals, especially quadratic ones, linking to matroid conjectures.

## Key findings

- Betti numbers are polynomial in the power k for studied ideals.
- Explicit formulas for Betti numbers of square-free monomial ideals.
- Analysis of Rees ideal generators, focusing on quadratic generation.

## Abstract

An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear powers if $I^k$ has a linear minimal free resolution, for all $k$. In this paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers. The Betti numbers are computed explicitly, as polynomials in $k$, for the ideal generated by all square free monomials of degree $d$, for $d=2, 3$ or $n-1$, and the product of all ideals generated by $s$ variables, for $s=n-1$ or $n-2$. We also study the generators of the Rees ideal, for ideals with linear powers. Especially, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This is related to a conjecture on matroids by White.