Related papers: The image of multilinear polynomials evaluated on …

We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the…

We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…

The single defining relation of the algebra of $SL_3\times SL_3$-invariants of triples of $3\times 3$ matrices is explicitly computed. Connections to some other prominent algebras of invariants are pointed out.

We study integrals over the triangle with vertices (1,0), (0,1), (1,1) that give linear combinations of multiple zeta values.

We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give…

We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.

Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak…

Let us consider a polynomial algebra in three variables equipped with an integer grading. We construct a system of group-generating automorphisms that preserve a given grading.

This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincar\'e from dimension two to dimension…

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all…

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…

After defining convex near-polygons, a formula enumerating the number of triangulations of such configurations is derived in terms of edge-polynomials. The paper describes also a transfer-matrix approach for computing quantities related to…

Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle…

The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many…

Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading…

We study the following problem in computer vision from the perspective of algebraic geometry: Using $m$ pinhole cameras we take $m$ pictures of a line in $\mathbb P^3$. This produces $m$ lines in $\mathbb P^2$ and the question is which…

We describe the T-ideal of identities and the T-space of central polynomials for the infinite dimensional unitary Grassmann algebra over a finite field.