arXiv subjects

Denis Ovchinnikov

Publications and source records attributed to Denis Ovchinnikov.

7 recordsLinked to original sources

arXiv2022

Studying the Diophantine problem in finitely generated rings and algebras via bilinear maps

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is Karp-reducible to the same problem in $R$. In several cases we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$, which implies that the Diophantine problem in $R$ is undecidable in this case. Otherwise, the ring $O$ is a ring of algebraic integers, and then the long-standing conjecture that $\mathbb{Z}$ is always interpretable by systems of equations in $O$ carries over to $R$. If true, it implies that the Diophantine problem in $R$ is also undecidable. Some of the classes of f.g. rings studied in this paper are the following: all associative, commutative, non-unitary rings (a similar statement for the unitary case was obtained by Eisentraeger); all possibly non-associative, non-commutative non-unitary rings that are f.g. as an abelian group; and several classes of f.g. non-commutative rings. Analogous statements are obtained for algebras over f.g. associative commutative unitary rings. Another contribution is the technique by which the aforementioned results are obtained: We show that given a bilinear map $f: A\times B \to C$ between f.g. abelian groups (or modules), under mild assumptions, there exists a certain ring (or algebra) $R$ with nice properties which is interpretable by systems of equations in the multi-sorted structure $(A,B,C;f)$. This result is not only relevant for rings and algebras, but also in other structures such as groups, as demonstrated previously by the authors.

math.RA↗

arXiv2021

The Diophantine problem in finitely generated commutative rings

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 field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is reducible to the same problem in $R$. If, in particular, $R$ has positive characteristic or, more generally, if $R$ has infinite rank, then we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$. This implies that the Diophantine problem in $R$ is undecidable in this case. In the remaining case where $R$ has finite rank and zero characteristic, we see that $O$ is a ring of algebraic integers, and then the long-standing conjecture that $\mathbb{Z}$ is always interpretable by systems of equations in a ring of algebraic integers carries over to $R$. If true, it implies that the Diophantine problem in $R$ is also undecidable. Thus, in this case the Diophantine problem in every infinite finitely generated commutative unitary ring is undecidable. The present is the first in a series of papers were we study the Diophantine problem in different types of rings and algebras.

math.NT↗

arXiv2020

Full rank presentations and nilpotent groups: structure, Diophantine problem, and genericity

We study finitely generated nilpotent groups $G$ given by full rank finite presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the deficiency $|A| - |R| $ is at least $2$ then the group $G$ is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations $\langle A \mid R\rangle$ is undecidable if $|A| - |R| \geq 2$ and decidable otherwise. We show that this class of groups is rather large since finite presentations asymptotically almost surely have full rank, so a random nilpotent group in the few relators model has a full rank presentation asymptotically almost surely. Full rank presentations give one a useful tool to approach random nilpotent groups and study their properties. Note, that the results above significantly improve our understanding of the Diophantine problem in finitely generated nilpotent groups: from a few special examples of groups with undecidable Diophantine problem we got to the place where we know that the Diophantine problem in all "typical" nilpotent groups is also undecidable.

math.GR↗

arXiv2020

Metabelian groups: full-rank presentations, randomness and Diophantine problems

We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that if $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\geq |A|$. We further prove that if $|R| \leq |A|-1$ then in any direct decomposition of $G$ all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

math.GR↗

arXiv2020

Diophantine problems in solvable groups

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory states that the Diophantine problem in any such $O$ is undecidable. If true this would imply that the Diophantine problem in any such $G$ is also undecidable. Furthermore, we show that for many particular groups $G$ as above, the ring $O$ is isomorphic to the ring of integers $\mathbb{Z}$, so the Diophantine problem in $G$ is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups $UT(n,\mathbb{Z}), n \geq 3$. Then we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups $GL(3,\mathbb{Z}), SL(3,\mathbb{Z}), T(3,\mathbb{Z})$.

math.GR↗

arXiv2017

Low-complexity computations for nilpotent subgroup problems

We solve the following algorithmic problems using TC0 circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal'cev coordinates the algorithms run in quartic time.

math.GR↗

arXiv2016

Random nilpotent groups, polycyclic presentations, and Diophantine problems

We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $τ_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \label{tau2pres_0}\langle A, C \mid [a_i, a_j]= \prod_t {\scriptstyle c_t^{\scriptscriptstyle λ_{t,i,j}}} \ (i< j), \ [A,C]=[C,C]=1\rangle,$ where $A=\{a_1, \dots, a_n\}$, and $C=\{c_1, \dots, c_m\}$. Hence, one may select a random $τ_2$-group $G$ by fixing $A$ and $C$, and then randomly choosing exponents $λ_{t,i,j}$ with $|λ_{t,i,j}|\leq \ell$, for some $\ell$. We prove that, if $m\geq n-1\geq 1$, then the following holds asymptotically almost surely, as $\ell\to \infty$: The ring of integers $\mathbb{Z}$ is e-definable in $G$, systems of equations over $\mathbb{Z}$ are reducible to systems over $G$ (and hence they are undecidable), the maximal ring of scalars of $G$ is $\mathbb{Z}$, $G$ is indecomposable as a direct product of non-abelian factors, and $Z(G)=\langle C \rangle$. If, additionally, $m \leq n(n-1)/2$, then $G$ is regular (i.e. $Z(G)\leq {\it Is}(G')$). This is not the case if $m > n(n-1)/2$. In the last section of the paper we introduce similar models of random polycyclic groups and random f.g. nilpotent groups of any nilpotency step, possibly with torsion. We quickly see, however, that the latter yields finite groups a.a.s.

math.GR↗