On the score sheets of a round-robin football tournament

TL;DR

This paper investigates the algebraic structure of score sheets in round-robin football tournaments, analyzing key invariants like the Hilbert basis and Hilbert series through theoretical and computational methods.

Contribution

It introduces a novel algebraic framework for understanding score sheets as an affine monoid and computes its fundamental invariants.

Findings

Determined the Hilbert basis of the score sheet monoid.

Computed the Hilbert series and multiplicity.

Provided insights into the algebraic structure of tournament score sheets.

Abstract

The set of (ordered) score sheets of a round-robin football tournament played between $n$ teams together with the pointwise addition has the structure of an affine monoid. In this paper we study (using both theoretical and computational methods) the most important invariants of this monoid, namely the Hilbert basis, the multiplicity, the Hilbert series and the Hilbert function.