There is no polynomial formula for the catenary and the tame degree of finitely generated monoids

TL;DR

The work addresses whether a polynomial formula can capture the catenary degree $\mathsf c(H)$ and the tame degree $\mathsf t(H)$ across broad classes of finitely generated monoids. By formalizing notions of polynomial and implicit polynomial formulas for invariants and employing a projective-geometry–based criterion together with generic presentations, the authors prove that no such polynomial description exists for $\mathsf c(H)$ or $\mathsf t(H)$ when the class includes all numerical monoids generated by three atoms. They construct an infinite family of three-generator numerical monoids and show the invariant can be written as the maximum of three distinct nonconstant linear forms, which, via the criterion, rules out any polynomial formula. The result implies a fundamental limitation on describing these invariants in terms of atoms, though elasticity remains amenable to an implicit polynomial description, and other invariants can be expressed via alternative algebraic data.

Abstract

In the last two decades there has been a wealth of results determining the precise value of the catenary degree and the tame degree. Mostly, however, only for very special classes of monoids and domains. In the present work we now show that there is no polynomial formula, neither for the catenary nor for the tame degree, which is valid for a sufficiently large class of finitely generated monoids.

Theorems & Definitions (5)

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• proof : Proof of Theorem \ref{['main']}