Newton complementary duals of $f$-ideals

TL;DR

This paper explores the properties of $f$-ideals and their Newton complementary duals, revealing a duality that extends previous results and offers a new perspective on the Kruskal-Katona theorem for simplicial complexes.

Contribution

It proves that an $f$-ideal's Newton complementary dual is also an $f$-ideal, broadening the class of $f$-ideals and providing an alternative formulation of the Kruskal-Katona theorem.

Findings

$f$-ideals are preserved under Newton complementary duality

Extended results to larger classes of $f$-ideals

Provided a new formulation of the Kruskal-Katona theorem

Abstract

A square-free monomial ideal $I$ of $k[x_1,\ldots,x_n]$ is said to be an $f$-ideal if the facet complex and non-face complex associated with $I$ have the same $f$-vector. We show that $I$ is an $f$-ideal if and only if its Newton complementary dual $\widehat{I}$ is also an $f$-ideal. Because of this duality, previous results about some classes of $f$-ideals can be extended to a much larger class of $f$-ideals. An interesting by-product of our work is an alternative formulation of the Kruskal-Katona theorem for $f$-vectors of simplicial complexes.