Year: 2025
Type: article
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.
Source: Journal of Symbolic Logic
Authors: NATALIA GARCIA-FRITZ, HECTOR PASTEN, THANASES PHEIDAS
Institutions: Pontificia Universidad Católica de Chile, University of Crete
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416129041
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