Related papers: On the typical rank of real binary forms

Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is…

All quasi-affine connected Generalized Dynkin Diagram with rank $= 4$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 4$ are also found.

Pick a sequence of uniform points on the $d$-dimensional sphere. Then, link the $n$th point to its closest one that arrives in the past. This constructs a labelled tree called the nearest neighbour tree on the $d$-dimensional sphere. These…

Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having…

In order to understand the parameter space of monic and centered complex polynomial vector fields of degree d in the complex plane, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci…

We classify all rational functions whose branching pattern above {0, 1, infinity} satisfy a certain regularity condition with precisely d=5 exceptions. This work is motivated by solving second order linear differential equations, with d=5…

We prove that any odd-dimensional modular category of rank at most 23 is pointed. We also show that an odd-dimensional modular category of rank 25 is either pointed, perfect, or equivalent to $\operatorname{Rep}(D^\omega(\mathbb Z_7\rtimes…

In a series of papers we investigated the following question: given a family $\calF$ of binary forms having nonzero discriminant and integer coefficients, for each $d\geqslant 3$, we estimate the number of integers $m$ with $|m|\leqslant N$…

In these notes we investigate the rings of real polynomials in four variables, which are invariant under the action of the reflectiongroups [3,4,3] and [3,3,5]. It is well known that they are rationally generated in degree 2,6,8,12 and…

All quasi-affine connected Generalized Dynkin Diagram with rank $= 3$ and $2$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 3$ and $2$ are also found.

It is known that degrees of basic relative invariants of homogeneous open convex cones of rank $r$ are less than or equal to $2^{r-1}$. In this article, we show that there exists a homogeneous cone of rank $r$ one of whose basic relative…

Given a binary form $F \in \mathbb{Z}[X, Y]$, we define its value set to be $\{F(x, y) : (x, y) \in \mathbb{Z}^2\}$. Let $F, G \in \mathbb{Z}[X, Y]$ be two binary forms of degree $d \geq 3$ and with non-zero discriminant. In a series of…

For each natural number $n$, we define a category whose objects are discriminant algebras in rank $n$, i.e. functorial means of attaching to each rank-$n$ algebra a quadratic algebra with the same discriminant. We show that the discriminant…

In this paper, we study the rank of matrices of bicomplex numbers. The relationship between rank, idempotent column rank and idempotent row rank is examined. Then, the solution of a system of equations in bicomplex space is presented using…

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…

In this note, we use a natural desingularization of the conormal variety of the variety of n x n symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.

We consider $p$-orientations, which are defined to be orientations of $d$-regular graphs such that every vertex either has in-degree $p$ or out-degree $p$. These generalise the orientations considered in Jaeger's conjecture, where $d=4p+1$.…

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees in which each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable…

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…