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NON-DIOPHANTINE SETS IN RINGS OF FUNCTIONS

Part of: Algebraic number theory: global fields Polynomials and matrices Number theory: Connections with logic

Published online by Cambridge University Press:  12 November 2025

NATALIA GARCIA-FRITZ ,
HECTOR PASTEN  and
THANASES PHEIDAS
NATALIA GARCIA-FRITZ
Affiliation:
DEPARTMENT OF MATHEMATICS PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE CHILE E-mail: natalia.garcia@uc.cl
HECTOR PASTEN*
Affiliation:
DEPARTMENT OF MATHEMATICS PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE CHILE E-mail: natalia.garcia@uc.cl
THANASES PHEIDAS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CRETE GREECE E-mail: pheidas@uoc.gr
*
E-mail: hector.pasten@uc.cl
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Abstract

Except for a limited number of cases, a complete classification of the Diophantine (i.e., positive existentially definable) sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine. For instance, we show that the relation of equality of degrees is not Diophantine over the field of complex rational functions in one variable and, in the same structure, we show that certain family of relations that approximates the valuation ring at infinity is not Diophantine either.

Keywords

Diophantine setspolynomialsfunction fields

MSC classification

Primary: 11U05: Decidability
Secondary: 11C08: Polynomials 11R58: Arithmetic theory of algebraic function fields

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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