Related papers: Generators, extremals and bases of max cones

Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define…

The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.

The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^(1/2)…

We obtain limit theorems for extreme residuals in linear regression model in the case of minimax estimation of parameters.

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The…

We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Grobner basis gives an algorithm for…

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group…

Let $K/k$ be a finite extension of a global field. Such an extension can be generated over $k$ by a single element. The aim of this article is to prove the existence of a "small" generator in the function field case. This answers the…

We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels.…

A generating set for a finite group $G$ is said to be minimal if no proper subset generates $G$, and $m(G)$ denotes the maximal size of a minimal generating set for $G$. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing…

We examine the extremal rays of the cone of dominant weights $(\mu, \widehat\mu)$ for groups $G\subseteq \widehat G$ for which there exists $N \gg0$ such that $$ \left(V(N\mu)\otimes V(N\widehat \mu)\right)^G\ne (0). $$ We exhibit formulas…

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the…

We prove the existence of self-dual tensor products for finite-dimensional convex cones and operator systems. This is a consequence of a more general result: Every cone system, which is contained in its dual, can be enlarged to a self-dual…

Let $\KX =K\langle X_1,\ldots ,X_n\rangle$ be the free algebra generated by $X=\{ X_1,\ldots ,X_n\}$ over a field $K$. It is shown that with respect to any weighted $\mathbb{N}$-gradation attached to $\KX$, minimal homogeneous generating…

We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We also…

Best possible bounds are established for families without s pairwise disjoint members and the more general problem for several families. The results are shown to apply several classical results.

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull dimension $d$, then any projective $R$-module of rank $n$ can be generated by $d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there…

In tropical geometry, there are several important classes of ideals and congruences such as tropical ideals, bend congruences, and the congruences of the form $\mathbf E(Z)$. Although they are analogues of the concept of ideals of rings, it…

We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then consider their minimal number of…