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October 3, 2025Open Access

A ruled residue theorem for function fields of hyperelliptic curves

Key Points

At most finitely many extensions of valuation exist for function fields of hyperelliptic curves.
Residue field extension can be transcendental but not ruled under certain conditions.
The research assumes residue characteristic is either zero or exceeds the degree of the hyperelliptic curve.
Valuation extensions reveal insights into the nature of residues in function fields.

Abstract

We study residually transcendental extensions of a valuation v on a field E to function fields of hyperelliptic curves over E. We show that v has at most finitely many extensions to the function field of a hyperelliptic curve over E, for which the residue field extension is transcendental but not ruled, assuming that the residue characteristic of v is either zero or greater than the degree of the hyperelliptic curve.

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Cite This Study

Gupta et al. (Mon,) studied this question.

synapsesocial.com/papers/68e040e5a99c246f578b2e7b https://doi.org/https://doi.org/10.48550/arxiv.2507.10365

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