Macaulay-like marked bases

TL;DR

This paper introduces a new class of marked bases over quasi-stable ideals in polynomial rings, demonstrating their geometric properties and providing algorithms for their explicit construction, extending Macaulay-like bases to more general settings.

Contribution

It defines marked bases over quasi-stable ideals, proves their scheme-theoretic and flatness properties, and develops algorithms for their explicit construction, generalizing Macaulay bases.

Findings

Marked bases form an affine scheme and are flat over the base.

For large enough m, marked bases form an open subset of a Hilbert scheme.

Algorithms for constructing marked bases are explicitly provided.

Abstract

We define marked sets and bases over a quasi-stable ideal $\mathfrak j$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of $\mathfrak j$ and a given integer $m$. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough $m$, is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.