• Corpus ID: 218571448

# A Note on Hilbert's "Geometric" Tenth Problem.

@article{Tyrrell2020ANO,
title={A Note on Hilbert's "Geometric" Tenth Problem.},
author={Brian Tyrrell},
journal={arXiv: Logic},
year={2020}
}
• B. Tyrrell
• Published 20 September 2019
• Mathematics
• arXiv: Logic
This paper explores the decidability of the existential theory of positive characteristic function fields in the "geometric" language of rings $\mathcal{L}_F = \{0, 1, +, \cdot, F\}$, with a unary predicate $F$ for nonconstant elements. In particular we are motivated by a question of Fehm on the decidability of $\mbox{Th}_{\exists}(\mathbb{F}_p(t); \mathcal{L}_F)$; equivalently, that of $\mbox{Th}_{\exists}(\mathbb{F}_p(t); \mathcal{L}_{\mbox{rings}})$ without parameters. We prove $\mbox{Th}_… ## References SHOWING 1-10 OF 35 REFERENCES On definitions of polynomials over function fields of positive characteristi We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let$\G_p$be an algebraic Definability of Frobenius orbits and a result on rational distance sets We prove that the first order theory of (possibly transcendental) meromorphic functions of positive characteristic $$p>2$$p>2 is undecidable. We also establish a negative solution to an analogue of Undecidability and definability for the theory of global fields We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse's Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly On Dipphantine definability and decidability in some rings of algebraic functions of characteristic 0 There exists a set W′ of K-primes such that Hilbert's Tenth Problem is not decidable over OK, and the set (W′ ∖ W)∪(W √ W′) is finite. Elliptic divisibility sequences and undecidable problems about rational points • Mathematics • 2004 Abstract Julia Robinson has given a first-order definition of the rational integers ℤ in the rational numbers ℚ by a formula (∀∃∀∃) (F = 0) where the ∀-quantifiers run over a total of 8 variables, Hilbert's Tenth Problem over Function Fields of Positive Characteristic Not Containing the Algebraic Closure of a Finite Field • Mathematics • 2013 We prove that the existential theory of any function field$K$of characteristic$p> 0\$ is undecidable in the language of rings provided that the constant field does not contain the algebraic closure
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