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arXiv · 1810.07582

A note on linear resolution and polymatroidal ideals

Amir Mafi·Dler Naderi
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Abstract

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.

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Amir Mafi, Dler Naderi. 2018-10-17. A note on linear resolution and polymatroidal ideals. https://arxiv.org/abs/1810.07582

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