Macaulay's theorem for vector-spread algebras

Abstract

Let $S=K[x_1,\dots ,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let $\mathbf{t}=(t_1,\dots ,t_{d-1})\in {\mathbb{Z}}_{\ge 0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. To a $\mathbf{t}$-spread ideal $I$ in $S$, we associate a unique $f_{\mathbf{t}}$-vector and we prove that if $I$ is $\mathbf{t}$-spread strongly stable, then there exists a unique $\mathbf{t}$-spread lex ideal which shares the same $f_{\mathbf{t}}$-vector of $I$ via the combinatorics of the $\mathbf{t}$-spread shadows of special sets of monomials of $S$. Moreover, we characterize the possible $f_{\mathbf{t}}$-vectors of $\mathbf{t}$-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal-Katona. Finally, we prove that among all $\mathbf{t}$-spread strongly stable ideals with the same $f_{\mathbf{t}}$-vector, the $\mathbf{t}$-spread lex ideals have the largest Betti numbers.

This is the final version of our paper, accepted for publication in the International Journal of Algebra and Computation