Related papers: The Conjugacy Problem in the Grigorchuk Group is p…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
For some Maltsev conditions $\Sigma$ it is enough to check if a finite algebra $\mathbf A$ satisfies $\Sigma$ locally on subsets of bounded size, in order to decide, whether $\mathbf A$ satisfies $\Sigma$ (globally). This local-global…
We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by…
We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in…
We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups. We describe polynomial-time algorithms for checking if two elements in such a semigroup are ~p conjugate and whether…
The conjugator length function of a finitely generated group $\Gamma$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $\Gamma$. We…
The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge…
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply…
Graph homomorphism has been studied intensively. Given an m x m symmetric matrix A, the graph homomorphism function is defined as \[Z_A (G) = \sum_{f:V->[m]} \prod_{(u,v)\in E} A_{f(u),f(v)}, \] where G = (V,E) is any undirected graph. The…
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential…
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous…
We prove that when a digraph $G$ has a Maltsev polymorphism, then $G$ also has a majority polymorphism. We consider the consequences of this result for the structure of Maltsev graphs and the complexity of the Constraint Satisfaction…
A subgroup $\Delta\leq \Gamma$ is commensurated if $|\Delta:\Delta\cap \gamma\Delta\gamma^{-1}|<\infty$ for all $\gamma\in \Gamma$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is…
We aim at enforcing hard constraints to impose a global structure on sequences generated from Markov models. In this report, we study the complexity of sampling Markov sequences under two classes of constraints: Binary Equalities and…
We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d + \lambda\omega$,…
We prove that each \omega-categorical, generically stable group is solvable-by-finite.
Let $X$ be a compact metrizable group and $\Gamma$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\Gamma)$ is surjunctive, i.e., every injective…
We investigate the descriptive set-theoretic complexity of the solvability of a Borel family of linear equations over a finite field. Answering a question of Thornton, we show that this problem is already hard, namely $\Sigma^1_2$-complete.…