Related papers: Further results on Hilbert's Tenth Problem

For a finite $\mathbb{Z}$-algebra $R$, i.e., for a $\mathbb{Z}$-algebra which is a finitely generated $\mathbb{Z}$-module, we assume that $R$ is explicitly given by a system of $\mathbb{Z}$-module generators $G$, its relation module ${\rm…

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…

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 $…

The problem of deciding, given a complex variety X, a point x in X, and a subvariety Z of X, whether there is an automorphism of X mapping x into Z is proved undecidable. Along the way, we prove the undecidability of a version of Hilbert's…

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder…

In the study of Hilbert schemes, the integer partition $\lambda$ helps researchers identify some geometric and combinatorial properties of the scheme in question. To aid researchers in extracting such information from a Hilbert polynomial,…

For each algebraic number $\alpha$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(\alpha)$ creates an extremal problem whose solution varies depending on the value of $t$. The second author studied the points $t$ at…

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…

The paper considers the Hilbert space $\hat{H}_r$ of real functions summable with the square $L^2(a,b)_r$ on any interval $\{(a,b)_r\}_{r=1}^{\infty}\in \mathbb{R}$. It is shown on the basis of the theorem on zeros of real orthogonal…

We study the computational complexity of fundamental problems over the $p$-adic numbers ${\mathbb Q}_p$ and the $p$-adic integers ${\mathbb Z}_p$. Gu\'epin, Haase, and Worrell proved that checking satisfiability of systems of linear…

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$ is a $2\times 2$ zero-trace…

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…

Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves…

Efficient algorithms for many problems in optimization and computational algebra often arise from casting them as systems of polynomial equations. Blum, Shub, and Smale formalized this as Hilbert's Nullstellensatz Problem $HN_R$: given…

In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.

We consider the graph $k$-colouring problem encoded as a set of polynomial equations in the standard way over $0/1$-valued variables. We prove that there are bounded-degree graphs that do not have legal $k$-colourings but for which the…

A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y) = a\cdot p_1^{z_1}\cdots p_v^{z_v}, \qquad \gcd(X,Y)=1$$ where $F$ be an irreducible homogeneous binary form of degree at least $3$ with integer coefficients, $a$ is a…

We study the asymptotics of singular values and singular functions of a Finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a…

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 consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated…