Related papers: Undecidability on Diophantine equations over $\mat…

In this paper we show that there is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{52})=0$ (with integer coefficients) over the Gaussian ring $\mathbb Z[i]$ is solvable.

Hilbert's Tenth Problem (HTP) asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring $\mathbb Z$ of the integers. This was finally solved by Matiyasevich…

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…

Hilbert's 10th problem, stated in modern terms, is: Find an algorithm that will, given $p \in \mathbb{Z}[x_1,\ldots,x_n]$ determine if there exists $a_1, a_2, \ldots, a_n \in \mathbb{Z}$ such that $p(a_1,\ldots,a_n)=0$. Davis, Putnam,…

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's…

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

One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution,…

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…

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p…

Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each $f\in\mathbb{Z}[X_{1},\dots,X_{n}]$, whether the diophantine equation $f(X_{1},...,X_{n})=0$ has a solution in R. The celebrated…

Let K be the function field of a variety of dimension at least 2 over an algebraically closed field of characteristic zero. Then Hilbert's Tenth Problem for K is undecidable. This generalizes the result by Kim and Roush from 1992 that…

Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for…

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural…

For K \subseteq C, let B_n(K)={(x_1,...,x_n) \in K^n: for each y_1,...,y_n \in K the conjunction (\forall i \in {1,...,n} (x_i=1 => y_i=1)) AND (\forall i,j,k \in {1,...,n} (x_i+x_j=x_k => y_i+y_j=y_k)) AND (\forall i,j,k \in {1,...,n}…

In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving that $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ such that…

In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\ldots, A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a…

This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the…

The analogue of Hilbert's tenth problem over $\mathbb{Q}$ asks for an algorithm to decide the existence of rational points in algebraic varieties over this field. This remains as one of the main open problems in the area of undecidability…

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 show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K,…