# Further results on Hilbert’s Tenth Problem

@article{Sun2020FurtherRO, title={Further results on Hilbert’s Tenth Problem}, author={Zhi-Wei Sun}, journal={Science China Mathematics}, year={2020}, pages={1-26} }

Hilbert’s Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring ℤ of integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over ℤ. We show that there is no algorithm to determine for any P ( z 1 ,…, z 9 ) ∈ ℤ[ z 1 ,…, z g ] whether the equation P ( z 1 ,…, z g ) = 0 has integral solutions with z 9 ⩾ 0. Consequently, there is…

## 14 Citations

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,…

$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with 32 unknowns

- Mathematics
- 2021

Let Z be the ring of integers. Hilbert’s Tenth Problem (HTP), the tenth one of his 23 famous mathematical problems presented in the 1900 ICM, asks for an algorithm to determine for any given…

Hilbert's Tenth Problem: Refinements and Variants

- Mathematics
- 2021

Hilbert’s 10th problem, stated in modern terms, is Find an algorithm that will, given p ∈ Z [ x 1 , . . . , x n ] , determine if there exists a 1 , . . . , a n ∈ Z such that p ( a 1 , . . . , a n ) =…

Formalizing a Diophantine Representation of the Set of Prime Numbers

- Mathematics, Computer ScienceArXiv
- 2022

This work shows that the exponential function is diophantine, together with the same properties for the binomial coefficient and factorial, in the set of prime numbers and its explicit representation using 10 variables, the smallest representation known today.

Diophantine equations: a systematic approach

- Mathematics
- 2021

This paper initiates a novel research direction in the theory of Diophantine equations: deﬁne an appro-priate version of the equation’s size, order all polynomial Diophantine equations starting from…

Mixed quantifier prefixes over Diophantine equations with integer variables

- Mathematics
- 2021

In this paper we study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that ∀ 2 ∃ 4 over Z is undecidable, that is, there is no algorithm to…

On exponential diophantine equations over $\mathbb Q$ with few unknowns

- Mathematics
- 2021

In this paper we obtain three undecidable results for exponential diophantine equations over the field Q of rational numbers. For example, we prove that there is no algorithm to decide the…

Semi-algebraic sets of f-vectors

- MathematicsIsrael Journal of Mathematics
- 2019

Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases…

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.

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