Related papers: A survey of local-global methods for Hilbert's Ten…
This paper initiates a novel research direction in the theory of Diophantine equations: define an appropriate version of the equation's size, order all polynomial Diophantine equations starting from the smallest ones, and then solve the…
Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$…
For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an enumeration operator, mapping each set $W$ of prime numbers to $HTP(\mathbb…
Rice's theorem states that no non-trivial semantic property of programs is decidable. Classical proofs proceed by reduction from the halting problem, invoking the law of excluded middle (LEM) twice: once through diagonalization, and once…
One of the main open problems in the context of extensions of Hilbert's tenth problem (HTP) is the case of the ring of complex entire functions in one variable. Our main result provides a step towards an answer: For every $\rho\ge 0$, we…
The Riemann hypothesis is one of the most famous unresolved problems in modern mathematics. The discussion here will present an overview of past methods that prove the Riemann hypothesis is a $\Pi_1^0$ sentence. We also end with some…
Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the…
This paper explores undecidability in theories 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…
We show that the decidability of an amplification of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive…
We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one…
To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth…
Polynome codes and code evaluation; arithmetical theory frames; $\mu$-recursive race for decision; decision correctness; decision termination; correct termination in theory $T = PR$ of Primitive Recursion; comparison with the negative…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shr\"odinger propagator…
Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops…
In this paper we complete B\"{u}chi's proof that there is no decision algorithm for the solubility in integers of arbitrary systems of diagonal quadratic form equations, by proving the assertion that whenever $x_1^2, \cdots, x_5^2$ are five…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…
We relate the decidability problem for BS with unordered cartesian product with Hilbert's Tenth problem and prove that BS with unordered cartesian product is NP-complete.
The massive Thirring model in the non-laboratory coordinates is considered by the Riemann-Hilbert approach. Existence of global solutions is shown for the cases of the associated Riemann-Hilbert problem without eigenvalues or resonances.…
We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0,…
In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that $\forall^2\exists^4$ over $\mathbb Z$ is…