Parametric standard basis, degree bound and local Hilbert-Samuel function

TL;DR

This paper develops a comprehensive framework for standard bases of polynomial ideals with parameters, providing degree bounds and stratification methods for local Hilbert-Samuel functions applicable to arbitrary monomial orders.

Contribution

It introduces a general study of parametric standard bases with arbitrary monomial orders, including explicit degree bounds and stratification techniques for Hilbert-Samuel functions.

Findings

Explicit upper bounds for standard basis degrees.

Quantitative bounds on possible Hilbert-Samuel functions.

Applicability to arbitrary monomial orders.

Abstract

We propose a general study of standard bases of polynomial ideals with parameters in the case where the monomial order is arbitrary. We give an application to the computation of the stratification by the local Hilbert-Samuel function. Moreover, we give an explicit upper bound for the degree of a standard basis for an arbitrary order and also for the number of the possible affine or local Hilbert-Samuel functions depending on the number of variables and the maximal degree of the given generators.