Hilbert's Tenth Problem for rational function fields over p-adic fields

2011/06/24 by Claudia Degroote, Degroote, Claudia, Jeroen Demeyer +1 · 1 citation
Mathematics · #11E04 #11U05 (Primary) 11S05 #12J10 (Secondary) #14H05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #advanced mathematical theories #math.LO #math.NT #msc:11E04 #msc:11S05 #msc:11U05 #msc:12J10 #msc:14H05

paper · pdf · doi:10.48550/arxiv.1106.4912

arxiv created 2011/06/24 · openalex publication_date 2011/06/24 · arxiv updated 2011/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a p-adic field (a finite extension of some Q_p) and let K(t) be the field of rational functions over K. We define a kind of quadratic reciprocity symbol for polynomials over K and apply it to prove isotropy for a certain class of quadratic forms over K(t). Using this result, we give an existential definition for the predicate "v_t(x) >= 0" in K(t). This implies undecidability of diophantine equations over K(t).

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• Existentially defining valuations in function fields over large fields

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