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

Altenbernd, Thomas and W\"ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the B\"uchi and Muller ones [1]. It was proved…

The logic of bunched implications (BI), introduced by O'Hearn and Pym (1999), has attracted significant attention due to its elegant proof calculus, varied semantics, and close connections to the propositional fragment of separation logic.…

The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a$, $b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a$, $b$,…

We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we…

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial…

We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining…

In this paper we show that Diophantine problem for quadratic equations in Baumslag-Solitar groups $BS(1,k)$ and in wreath products $A \wr \mathbb{Z}$, where $A$ is a finitely generated abelian group and $\mathbb{Z}$ is an infinite cyclic…

A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible…

In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is…

For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable.…

The satisfiability problem for multilevel syllogistic extended with the Cartesian product operator (MLSC) is a long-standing open problem in computable set theory. For long, it was not excluded that such a problem were undecidable, due to…

In this paper, we consider iterative propositional calculi, which are finite sets of propositional formulas together with the rules of modus ponens and weak substitution (when formula being substituted must be already inferred). We…

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 $…

We establish the undecidability of conditional affine information inequalities, the undecidability of the conditional independence implication problem with a constraint that one random variable is binary, and the undecidability of the…

We prove that for each odd prime p, positive integer alpha, and non-negative integers beta and gamma, the Diophantine equation X^{2N} + 2^{2 alpha} 5^{2 beta} p^{2 gamma} = Z^5 has no solution with X, Z, N in Z^+, N > 1, and gcd(X,Z) = 1.

In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$…

For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is…

We give a survey on some results covering the last 60 years concerning Je\'smanowicz' conjecture. Moreover, we conclude the survey with a new result by showing that the special Diophantine equation $$(20k)^x+(99k)^y=(101k)^z$$ has no…

It is undecidable whether the language recognized by a probabilistic finite automaton is empty. Several other undecidability results, in particular regarding problems about matrix products, are based on this important theorem. We present…

We consider the thesis that an arithmetical relation, which holds for any, given, assignment of natural numbers to its free variables, is Turing-decidable if, and only if, it is the standard representation of a PA-provable formula. We show…