Hilbert's Tenth Problem for fields of rational functions over finite fields

1991 ◽  
Vol 103 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Thanases Pheidas
Keyword(s):  
Finite Fields ◽  
Rational Functions ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem

A Note on Hilbert's Tenth Problem

1960 ◽  
Vol 3 (2) ◽  
pp. 153-156
Author(s):  
Z. A. Melzak
Keyword(s):  
Decision Procedure ◽  
Rational Functions ◽  
Single Variable ◽  
Hilbert's Tenth Problem ◽  
Integer Coefficients ◽  
Arbitrary Polynomial ◽  
Hilbert’S Tenth Problem ◽  
Derivatives Of

The tenth problem on Hilbert's well known list [1] is the following.(H 10) For an arbitrary polynomial P = P(x1,x2,…,xn) with integer coefficients to determine whether or not the equation P = 0 has a solution in integers.By 'integers' we always mean 'rational integers'. The problem (H 10) is still unsolved but it appears likely that no decision procedure exists; in this connection see [2]. It will be shown here that (H 10) is equivalent to deciding whether or not every member of a certain given countable sec of rational functions of a single variable x is absolutely monotonie. We recall that f(x) is absolutely monotonie in I if f(x) possesses non-negative derivatives of all orders at every x ∊ I.

An analogue of Hilbert's tenth problem for p-adic entire functions

1995 ◽  
Vol 60 (4) ◽  
pp. 1301-1309 ◽  
Author(s):  
Leonard Lipshitz ◽  
Thanases Pheidas
Keyword(s):  
Entire Functions ◽  
Algebraic Closure ◽  
Rational Functions ◽  
Open Unit ◽  
Closed Field ◽  
Closed Disk ◽  
Hilbert's Tenth Problem ◽  
First Order Theory ◽  
Hilbert’S Tenth Problem ◽  
Language L

Let p be a fixed prime integer, other than 2, Qp the field of p-adic numbers, and Ωp the completion of the algebraic closure of Qp. Let Rp be the ring of entire functions in one variable t over Ωp; that is, Rp is the ring of functions f: Ωp → Qp such that f(t) is given by a power series around 0, of infinite radius of convergence:and where ∣a∣p is the p-adic norm of a in Ωp. We prove:Theorem A. The positive existential theory of Rp in the language L = {0, 1, t, +, ·} is undecidable.Theorem A gives a negative answer to the analogue of Hilbert's tenth problem for Rp in the language L. Related results include those of [2] where it is shown that the first-order theory of entire functions on the complex plane is undecidable and the similar result for analytic functions on the open unit disk (this is due to Denef and Gromov, communicated to us by Cherlin and is as of now unpublished).It would be desirable to have a similar result in the language which, instead of the variable t, has a predicate for the transcendental (that is, nonconstant) elements of Rp. A related problem is the similar problem for meromorphic functions on the real or p-adic plane or on the unit open or closed disk. These problems seem for the moment rather hard in view of the fact that the analogue of Hilbert's Tenth Problem for the field of rational functions over the complex numbers (or any algebraically closed field of characteristic zero) is an open problem.

Hilbert's Tenth Problem

2021 ◽  
Vol 52 (2) ◽  
pp. 36-44
Author(s):  
William Gasarch
Keyword(s):  
Short Version ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem ◽  
Complete History

This column is a short version of a long version of an article based on a blog. What? I give the complete history.

Hilbert's tenth problem

1975 ◽  
Vol 3 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Yu. I. Manin
Keyword(s):  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem
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