Related papers: Hilbert's Tenth Problem and Mazur's Conjectures in…

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.

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers…

We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring $R$ we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite…

Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under…

Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical…

Diophantine subsets of $\mathbb{Z}$ play a key role in the negative answer to Hilbert's tenth problem. The definition of diophantine set generalizes in several ways to other commutative rings. We compare these definitions. Along the way, we…

This expository article covers the recent developments surrounding Hilbert's tenth problem for finitely generated rings. We start by recounting the history of Hilbert's tenth problem over the integers, which was resolved negatively by…

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 operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$,…

We consider matrix problems in Hilbert spaces (orthoscalar representations of quivers and posets). A criterion of tameness of the problem of classification of indecomposable orthoscalar representations of a quiver is given.

We interpret a counterexample to Hilbert's 14th problem by S. Kuroda geometrically in two ways: As ring of regular functions on a smooth rational quasiprojective variety over any field K of characteristic 0, and, in the special case where K…

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the…

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,…

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…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is…

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…

Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values…

Let $K= \mathbf{Q}(\sqrt{d})$ be a quadratic field and $\mathcal{O}_{K}$ be its ring of integers. We study the solvability of the Diophantine equation $r + s + t = rst = 2$ in $\mathcal{O}_{K}$. We prove that except for $d= -7, -1, 17$ and…

Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove…

The paper deals with the {\it infinitesimal Hilbert 16th problem}: to find an upper estimate of the number of zeros of an Abelian integral regarded as a function of a parameter. In more details, consider a real polynomial $ H$ of degree $…