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Keywords:
Abhyankar's construction; semiring; semifield; finitely generated; additively idempotent
Abhyankar's construction; semiring; semifield; finitely generated; additively idempotent
Summary:
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 cannot 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.
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