The Gr\"obner basis for powers of a general linear form in a monomial complete intersection

TL;DR

This paper provides an explicit combinatorial description of Gr"obner bases for powers of a general linear form in monomial complete intersections, linking algebraic properties to lattice paths and classical combinatorial sequences.

Contribution

It introduces a new combinatorial approach to describe Gr"obner bases and proves the strong Lefschetz property for these ideals over characteristic zero fields.

Findings

Explicit Gr"obner bases for these ideals under any term order.

Connection of basis element counts to Catalan, Motzkin, and Riordan numbers.

New proof of the strong Lefschetz property in characteristic zero.

Abstract

We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gr\"obner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic $p$. Furthermore, we show that the number of Gr\"obner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics.