Related papers: The Cayley-Hamilton theorem for supermatrices

We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

Let A be a subspace arrangement and let chi(A,t) be the characteristic polynomial of its intersection lattice L(A). We show that if the subspaces in A are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t) counts a…

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by…

Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The…

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The object of this work is the search…

Starting from a mistake done by a student, we discover an unexpected method of finding both eigenvectors for a $2\times2$ matrix with distinct eigenvalues in a single computation. We discuss a connection with the Cayley-Hamilton theorem,…

We give a criterion for a quasi-ordinary polynomial to be irreducible. The criterion is based on the notion of approximate roots and that of generalized Newton polygons.

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from…

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type $A$ (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to…

In this paper we use the power of the outer exponential $\Lambda^B$ of a bivector $B$ to see the so-called invariant decomposition from a different perspective. This is deeply connected with the eigenvalues for the adjoint action of $B$, a…

In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice polynomial…

Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…

As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to…

In the present paper, we provide a cohomology group as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation,…

In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where…

This article provides a definition of a subdifferential for continuous functions based on homological considerations. We show that it satisfies all the requirement for a good notion of subdifferential. Moreover, we prove sublinearity, a…

The super-algebraic structure of a generalized version of the Jaynes-Cummings model is investigated. We find that a Z2 graded extension of the so(2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the…

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as…

A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices,…