scholarly journals Hilbert's Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field

2017 ◽  
Vol 19 (7) ◽  
pp. 2103-2138
Author(s):  
Kirsten Eisenträger ◽  
Alexandra Shlapentokh
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Function Fields ◽  
Algebraic Closure ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem

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functions on the open unit disk (this is due to Denef and Gromov, communicated to us by Cherlin and is as of now unpublished).It would be desirable to have a similar result in the language which, instead of the variable t, has a predicate for the transcendental (that is, nonconstant)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.

scholarly journals Minimal Models and Abundance for Positive Characteristic Log Surfaces

2014 ◽  
Vol 216 ◽  
pp. 1-70 ◽  
Author(s):  
Hiromu Tanaka
Keyword(s):  
Finite Field ◽  
Minimal Model ◽  
Positive Characteristic ◽  
Algebraic Closure ◽  
Model Program ◽  
Minimal Models ◽  
Base Field ◽  
Singular Surfaces ◽  
Minimal Model Program ◽  
Log Canonical

AbstractWe discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

scholarly journals The distribution of the residues of a quartic polynomial

1967 ◽  
Vol 8 (2) ◽  
pp. 67-88 ◽  
Author(s):  
K. McCann ◽  
K. S. Williams
Keyword(s):  
Finite Field ◽  
Finite Number ◽  
Galois Group ◽  
Riemann Hypothesis ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Closure ◽  
Quartic Polynomial ◽  
Algebraic Function Fields ◽  
Image Position

Let f(x) denote a polynomial of degree d defined over a finite field k with q = pnelements. B. J. Birch and H. P. F. Swinnerton-Dyer [1] have estimated the number N(f) of distinct values of y in k for which at least one of the roots ofis in k. They prove, using A. Weil's deep results [12] (that is, results depending on the Riemann hypothesis for algebraic function fields over a finite field) on the number of points on a finite number of curves, thatwhere λ is a certain constant and the constant implied by the O-symbol depends only on d. In fact, if G(f) denotes the Galois group of the equation (1.1) over k(y) and G+(f) its Galois group over k+(y), where k+ is the algebraic closure of k, then it is shown that λ depends only on G(f), G+(f) and d. It is pointed out that “in general”

A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

2021 ◽  
pp. 2250189
Author(s):  
R. Toledano
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Finite Subset ◽  
Function Fields ◽  
Algebraic Closure ◽  
Value Set ◽  
Towers Of Function Fields ◽  
Value Sets

In this paper, we introduce the notions of [Formula: see text]-polynomial and [Formula: see text]-minimal value set polynomial where [Formula: see text] is a polynomial over a finite field [Formula: see text] and [Formula: see text] is a finite subset of an algebraic closure of [Formula: see text]. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are [Formula: see text]-minimal value set polynomials for [Formula: see text], whose [Formula: see text]-value sets can be explicitly computed in terms of the monomial [Formula: see text].

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