Newton Complementary Duals of -Ideals

Abstract

AbstractA 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.