相关论文: Polynomial identities for hypermatrices

We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is…

We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an…

Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper,…

We revisit the classical theorem by Cayley and Hamilton, "{\em each endomorphism is a root of its own characteristic polynomial}", from the point of view of {\em Hasse--Schmidt derivations on an exterior algebra}

Some integral properties of Jack polynomials, hypergeometric functions and invariant polynomials are studied for real normed division algebras.

In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…

A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain…

We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity {\em not} involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a…

Starting from the free field realization of Kac-Moody Lie algebra, we define a generalized Yang-Yang function. Then for the Lie algebra of type $A_{n}$, we derive braiding and fusion matrix by braiding the thimble from the generalized…

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…

We establish an analogue of the fundamental theorem of algebra for polynomial matrix equations, in which the matrices-coefficients and unknown matrix are assumed to be circulant matrices.

If A is an n \times n matrix over a ring R satisfying the polynomial identity [x,y][u,v]=0, then an invariant Cayley-Hamilton identity of the form \Sigma A^{i}c_{i,j}A^{j}=0 with c_{i,j}\in R and c_{n,n}=(n!)^2 holds for A.

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-$t$-Macdonald polynomials of two rows.

This is a survey of results obtained jointly with E. Aljadeff and published in Adv. Math. 218 (2008), 1453-1495. We explain how to set up a theory of polynomial identities for comodule algebras over a Hopf algebra, and concentrate on the…

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain $R$. By exploiting a connection with symmetric determinantal representations, we characterize the image of the…

In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci…

We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to…

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…