Related papers: Diophantine Problem in Some Metabelian Groups
In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under…
Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for…
We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…
The Donald--Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and for the classical Weyl groups (whose rational group…
Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that…
In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i z_i = 1$ (and its variants) over the class of finite metabelian groups…
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…
In this paper we provide criteria for the insolvability of the Diophantine equation $x^2+D=y^n$. This result is then used to determine the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$. We also determine some criteria for the…
We show that it is undecidable whether a system of linear equations over the Laurent polynomial ring $\mathbb{Z}[X^{\pm}]$ admit solutions where a specified subset of variables take value in the set of monomials $\{X^z \mid z \in…
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.
We find algebraic conditions on a group equivalent to the position of its Diophantine problem in the Chomsky Hierarchy. In particular, we prove that a finitely generated group has a context-free Diophantine problem if and only if it is…
For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine…
We prove the A-theoretic Isomorphism Conjecture with coefficients and finite wreath products for solvable groups.
Friedl and L\"oh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to a virtually cyclic group, or to the direct product of an abelian and a finite group, is decidable. Here we…
We prove that the isomorphism problem is decidable for generalized Baumslag-Solitar (GBS) groups with one quasi-conjugacy class and full support gaps. In order to do so we introduce a family of invariants that fully characterize the…
We prove that metabelian Baumslag$-$Solitar group $BS(1,k)$, $k>1$, is (strongly) regularly bi-interpretable with the ring of integers $\mathbb{Z}$, and describe in algebraic terms all groups that are elementarily equivalent to $BS(1,k)$.
We prove that the Word problem in the Baumslag group G(1,2) which has a non-elementary Dehn function is decidable in polynomial time.
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}…
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry…
We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big…