International Journal of Algebra and Computation

We solve the subgroup intersection problem (SIP) for any RAAG $G$ of Droms type (i.e. with defining graph not containing induced squares or paths of length $3$): there is an algorithm which, given finite sets of generators for two subgroups $H,K\le G$, decides whether $H\cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H\cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. ${𝔽}_2\times {𝔽}_2$) even have unsolvable SIP.

The degree of commutativity of a group $G$ measures the probability of choosing two elements in $G$ which commute. There are many results studying this for finite groups. In [Y. Antolín, A. Martino and E. Ventura, Degree of commutativity of infinite groups, Proc. Amer. Math. Soc. 145 (2017) 479–485, MR 3577854], this was generalized to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form $\mathbb{Z}\wr \mathbb{Z}$ and $F\wr \mathbb{Z}$, where $F$ is any finite group.

Let $G$ be a finite group and $I_G(\sigma )$ be the set of the elements $x$ of $G$ such that $\sigma (x)=x^{-1}$ where $\sigma \in Aut(G)$. In this paper, we give strong restrictions on the structure of the quotients $\frac{G}{O_2(G)}$ where $G$ is a finite group with an automorphism $\sigma$ such that $|I_G(\sigma )|\ge \frac{\left|G\right|}{3}$ and $O_2(G)$ is the largest normal $2$-subgroup of $G$.

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to ${𝒜}$-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic $0$. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover, we provide a construction of minimal linear representations for the inverse of nonzero elements.

Böröczky configurations of lines have been recently considered in connection with the problem of the containment between symbolic and ordinary powers of ideals. Here, we describe parameter families of Böröczky configurations of $13$, $14$, $16$, $18$ and $24$ lines and investigate rational points of these parameter spaces.

We show that two-dimensional systolic complexes are quasi-isometric to quadric complexes with flat intervals. We use this fact along with the weight function of Brodzki, Campbell, Guentner, Niblo and Wright [J. Brodzki, S. J. Campbell, E. Guentner, G. A. Niblo and N. J. Wright, Property A and $\mathrm{\text{CAT}}(0)$ cube complexes, J. Funct. Anal. 256(5) (2009) 1408–1431] to prove that two-dimensional systolic complexes satisfy property A.

One studies certain degenerations of the generic square matrix over a field $k$ along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these derivatives, the Hessian matrix and the ideal of the submaximal minors of the matrix. The main tool comes from commutative algebra, with emphasis on ideal theory and syzygy theory. The structure of the polar map is completely identified and the main properties of the ideal of submaximal minors are determined. Cases where the degenerated determinant has non-vanishing Hessian determinant show that the former is a factor of the latter with the (Segre) expected multiplicity, a problem envisaged by Landsberg–Manivel–Ressayre by geometric means. Another byproduct is an affirmative answer to a question of F. Russo concerning the codimension in the polar image of the dual variety to a hypersurface.

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a “hydra phenomenon” gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here, we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.