Semifields and a theorem of Abhyankar

TL;DR

This paper discusses Abhyankar's theorem on fields as homomorphic images of polynomial rings and explores its implications for the structure of finitely generated semifields, proposing a conjecture linking these areas.

Contribution

It introduces a conjecture suggesting that Abhyankar's theorem cannot be significantly improved and connects this to a known conjecture on semifields' additive idempotence.

Findings

Conjecture implies a well-known open problem in semifield theory

Supports the belief that Abhyankar's theorem is near optimal

Highlights the relationship between polynomial rings and semifield structures

Abstract

Abhyankar proved that every field of finite transcendence degree over $\mathbb{Q}$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $\mathbb{Z}[T_1, \dots, T_n]$ (for some $n$ depending on the field). We conjecture that his result can not be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.