Related papers: New Facts about Berezinians

Symmetric polynomials of the roots of a polynomial can be written as polynomials of the coefficients, and by applying this to the characteristic polynomial we can write a symmetric polynomial of the eigenvalues $a_{i}$ of an $n\times n$…

We consider measures supported on the bi-circle and review the recurrence relations satisfied by the orthogonal polynomials associated with these measures constructed using the lexicographical or reverse lexicographical ordering. New…

We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we…

We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…

New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation…

We describe the symbolic calculus of operators on the unit sphere in the complex n-space $\mathbb C^n$ defined by the Berezin quantization. In particular, we derive a explicit formula for the composition of Berezin symbol and with that a…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

In this paper we consider pentadiagonal $(n+1)\times(n+1)$ matrices with two subdiagonals and two superdiagonals at distances $k$ and $2k$ from the main diagonal where $1\le k<2k\le n$. We give an explicit formula for their determinants and…

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…

As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given…

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…

We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive…

We find expressions of the polynomials defining the dual varieties of Grassmannians $Gr(3,9)$ and $Gr(4,8)$ both in terms of the fundamental invariants and in terms of a generic semi-simple element. We project the polynomial defining the…

We give an elementary combinatorial proof of Bass's determinant formula for the zeta function of a finite regular graph. This is done by expressing the number of non-backtracking cycles of a given length in terms of Chebychev polynomials in…

We give an explicit formula for the discriminant $\Delta_f (x)$ of the quadrinomials of the form $f (x)=x^n+ t (x^2+ax+b)$. The proof uses Bezoutians of polynomials.

In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write…

The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The…

We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…

We give a brief review of our approach to the quantization of superstrings. New is a covariant derivation of the measure at tree level and a path integral formula for this measure.

A class of derivatives is defined for the pseudo determinant $Det(A)$ of a Hermitian matrix $A$. This class is shown to be non-empty and to have a unique, canonical member $\mathbf{\nabla Det}(A)=Det(A)A^+$, where $A^+$ is the Moore-Penrose…