# Undecidable diophantine equations

@article{Jones1980UndecidableDE, title={Undecidable diophantine equations}, author={James P. Jones}, journal={Bulletin of the American Mathematical Society}, year={1980}, volume={3}, pages={859-862} }

In 1900 Hubert asked for an algorithm to decide the solvability of all diophantine equations, P(x1, . . . , xv) = 0, where P is a polynomial with integer coefficients. In special cases of Hilbert's tenth problem, such algorithms are known. Siegel [7] gives an algorithm for all polynomials P(xx, . . . , xv) of degree < 2. From the work of A. Baker [1] we know that there is also a decision procedure for the case of homogeneous polynomials in two variables, P(x, y) = c. The first steps toward the…

## 27 Citations

Universal diophantine equation

- MathematicsJournal of Symbolic Logic
- 1982

Matijasevic's theorem implies the existence of a diophantine equation U such that for all x and v, x ∈ W v is also recursively enumerable, and the nonexistence of such an algorithm follows immediately from theexistence of r.e. nonrecursive sets.

1 Hilbert ’ s Tenth Problem

- Mathematics
- 2021

Hilbert’s 10th problem, stated in modern terms, is Find an algorithm that will, given p ∈ Z[x1, . . . , xn], determine if there exists a1, . . . , an ∈ Z such that p(a1, . . . , an) = 0. Davis,…

Extensions of Hilbert's tenth problem

- MathematicsJournal of Symbolic Logic
- 1994

The present article is an attempt to bridge the gap between the researchers that work in the areas adjacent to Hilbert's Tenth Problem (for short, HTP), mainly, number theory and mathematical logic.…

The Fixed Point Problem for General and for Linear SRL Programs is Undecidable

- Computer ScienceICTCS
- 2018

It is shown that the existence of xed points in SRL is undecidable and complete in Σ 1 and that the problem of deciding if there is a tuple of initial register values of a given program P that remains unaltered after the execution of P.

A universal differential equation

- Mathematics, Philosophy
- 1981

REMARK 1. From the proof, it will be clear that we can in addition ensure that y(tj) — y(tj) for any sequence (tj) of distinct real numbers such that \tj\ —* °° as ƒ —> oo. REMARK 2. We may moreover…

POINTS AND SUBSPACES OF SMALL HEIGHT IN QUADRATIC AND LINEAR SPACES

- Mathematics
- 2012

We investigate a few related questions on search bounds via height for rational points over global fields in quadratic and linear spaces. Specifically, let K be a global field or Q and N ≥ 2 an…

Finding Zeros of Rational Quadratic Forms

- Mathematics
- 2014

In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski…

Representing integers by multilinear polynomials

- Mathematics
- 2020

Let
$$F(\varvec{x})$$
be a homogeneous polynomial in
$$n \ge 1$$
variables of degree
$$1 \le d \le n$$
with integer coefficients so that its degree in every variable is equal to 1. We give…

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