Distinct monomial orders with same induced orderings

TL;DR

This paper investigates the uniqueness of certain monomial orders in polynomial rings, proving that some are uniquely determined by their induced orderings for dimensions four and higher, while others are not.

Contribution

It establishes conditions under which monomial orders are uniquely determined by their induced orderings and provides explicit examples of non-uniqueness for dimensions four and above.

Findings

Lexicographic, degree lexicographic, and degree reverse lexicographic orders are uniquely determined by induced orderings for n ≥ 4.

There exist monomial orders that are not uniquely determined by their induced orderings for n ≥ 4.

Explicit examples of non-unique monomial orders are provided for each n ≥ 4.

Abstract

We prove that the lexicographic, degree lexicographic and the degree reverse lexicographic orders for monomials in $R_n=K[X_1,...X_n]$ are uniquely determined by their induced orderings, (i.e. their restrictions to $R_{n,i}=K[X_1,..., \hat{X_i}, .., X_n]$), when $n \geq 4$. We also show that for any $n \geq 4$ there are monomial orders that are not uniquely determined by their induced orderings, and provide examples of these orders for each $n$.