Volume 33 (2025), Issue 1


1. Weak polynomial identities of small degree for the Weyl algebra

Artem Lopatin ; Carlos Arturo Rodriguez Palma ; Liming Tang.
In this paper we investigate weak polynomial identities for the Weyl algebra $\mathsf{A}_1$ over an infinite field of arbitrary characteristic. Namely, we describe weak polynomial identities of the minimal degree, which is three, and of degrees 4 and 5. We also describe weak polynomial identities is two variables.

2. A $\delta$-first Whitehead Lemma for Jordan algebras

Arezoo Zohrabi ; Pasha Zusmanovich.
We compute $\delta$-derivations of simple Jordan algebras with values in irreducible bimodules. They turn out to be either ordinary derivations ($\delta = 1$), or scalar multiples of the identity map ($\delta = \frac 12$). This can be considered as a generalization of the "First Whitehead Lemma" for Jordan algebras which claims that all such ordinary derivations are inner. The proof amounts to simple calculations in matrix algebras, or, in the case of Jordan algebras of a symmetric bilinear form, to more elaborated calculations in Clifford algebras.

3. On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices

Diogo Diniz ; Felipe Yukihide Yasumura.
We study the algebra of upper triangular matrices endowed with a group grading and a homogeneous involution over an infinite field. We compute the asymptotic behaviour of its (graded) star-codimension sequence. It turns out that the asymptotic growth of the sequence is independent of the grading and the involution under consideration, depending solely on the size of the matrix algebra. This independence of the group grading also applies to the graded codimension sequence of the associative algebra of upper triangular matrices.

4. Symmetry groups and deformations of sums of exponentials

Florian Pausinger ; David Petrecca.
We study the symmetry groups and winding numbers of planar curves obtained as images of weighted sums of exponentials. More generally, we study the image of the complex unit circle under a finite or infinite Laurent series using a particular parametrization of the circle. We generalize various previous results on such sums of exponentials and relate them to other classes of curves present in the literature. Moreover, we consider the evolution under the wave equation of such curves for the case of binomials. Interestingly, our methods provide a unified and systematic way of constructing curves with prescribed properties, such as the number of cusps, the number of intersection points or the winding number.

5. Separating symmetric polynomials over finite fields

Artem Lopatin ; Pedro Antonio Muniz Martins ; Lael Viana Lima.
The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb F}_q$ the set $S(n)$ is not a minimal separating set for symmetric polynomials in general. We determined when $S(n)$ is a minimal separating set for the algebra of symmetric polynomials having the least possible number of elements.

6. Constructions of well-rounded algebraic lattices over odd prime degree cyclic number fields

Robson Ricardo de Araujo ; Antonio Aparecido de Andrade ; Trajano Pires da Nóbrega Neto ; Jéfferson Luiz Rocha Bastos.
Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent vectors in its set of minimal vectors. Both classes of lattices have been applied for signal transmission in some channels, such as wiretap channels. Recently, some advances have been made in the search for well-rounded lattices that can be realized as algebraic lattices. Moreover, some works have been published studying algebraic lattices obtained from modules in cyclic number fields of odd prime degree $p$. In this work, we generalize some results of a recent work of Tran et al. and we provide new constructions of well-rounded algebraic lattices from a certain family of modules in the ring of integers of each of these fields when $p$ is ramified in its extension over the field of rational numbers.

7. On the shape of the connected components of the complement of two-dimensional Brownian random interlacements

Orphée Collin ; Serguei Popov.
We study the limiting shape of the connected components of the vacant set of two-dimensional Brownian random interlacements: we prove that the connected component around $x$ is close in distribution to a rescaled \emph{Brownian amoeba} in the regime when the distance from $x\in\mathbb{C}$ to the closest trajectory is small (which, in particular, includes the cases $x\to\infty$ with fixed intensity parameter $\alpha$, and $\alpha\to\infty$ with fixed $x$). We also obtain a new family of martingales built on the conditioned Brownian motion, which may be of independent interest.