Finite Sets of Affine Points with Unique Associated Monomial Order Quotient Bases

TL;DR

This paper studies zero-dimensional ideals with unique monomial quotient bases, showing that Cartesian sets have this property and exploring their relationship with such point sets, relevant to interpolation and coding theory.

Contribution

It characterizes zero-dimensional ideals with unique monomial quotient bases and establishes that Cartesian sets possess this property, linking geometric configurations to algebraic structures.

Findings

Vanishing ideals of Cartesian sets have unique monomial quotient bases

Cartesian sets are characterized by their unique quotient bases

The relation between Cartesian sets and point sets with unique bases is clarified

Abstract

The quotient bases for zero-dimensional ideals are often of interest in the investigation of multivariate polynomial interpolation, algebraic coding theory, and computational molecular biology, etc. In this paper, we discuss the properties of zero-dimensional ideals with unique monomial quotient bases, and verify that the vanishing ideals of Cartesian sets have unique monomial quotient bases. Furthermore, we reveal the relation between Cartesian sets and the point sets with unique associated monomial quotient bases.