A combinatorial proof of the Eisenbud-Goto conjecture for monomial curves and some simplicial semigroup rings

TL;DR

This paper provides a combinatorial proof of the Eisenbud-Goto conjecture specifically for monomial curves and extends the result to some simplicial affine semigroup rings, advancing understanding in algebraic geometry.

Contribution

It introduces a purely combinatorial proof for the Eisenbud-Goto conjecture in the case of monomial curves and verifies the conjecture for certain simplicial semigroup rings.

Findings

Proof of the Eisenbud-Goto conjecture for monomial curves

Verification of the conjecture for specific simplicial affine semigroup rings

Advancement in combinatorial methods in algebraic geometry

Abstract

We will give a pure combinatorial proof of the Eisenbud-Goto conjecture for arbitrary monomial curves. Moreover, we will show that the conjecture holds for certain simplicial affine semigroup rings.