Unmixed polymatroidal ideals

TL;DR

The paper addresses the problem of classifying unmixed polymatroidal ideals in a polynomial ring $R=K[x_1, ablaots,x_n]$. It provides a complete combinatorial/algebraic description: (i) a matroidal unmixed ideal of degree $d$ is the edge ideal of a complete $d$-uniform $m$-partite hypergraph that is $k$-balanced, and (ii) a general polymatroidal ideal of degree $d$ is unmixed if and only if it has one of a few structured forms, including $I=rak{m}^d$, a product of equal-height prime powers with disjoint generators, or a mixture of such primes with an unmixed matroidal factor $J$ satisfying certain sum-degrees $= abla(d)$. The degree-$2$ case is handled via a decomposition into unmixed matroidal ideals or $I=rak{m}^2$, and the results connect to Veronese-type ideals and CM properties. The authors develop a framework based on colon ideals and support considerations, providing explicit constructive descriptions and examples, thereby delivering a complete classification of unmixed polymatroidal ideals and enriching the understanding of their combinatorial underpinnings.

Abstract

Let $R=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I$ be a polymatroidal ideal of $R$. In this paper, we provide a comprehensive classification of all unmixed polymatroidal ideals. This work addresses a question raised by Herzog and Hibi in [10]

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• Proposition 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6