Related papers: Polynomials that preserve nonnegative monomial mat…

In this paper, we give some results on closed polynomials and factorially closed polynomial in $n$ variables. In particular, we give a characterization of factorially closed polynomials in $n$ variables over an algebraically closed field…

We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.

This paper is intended to give closed formulae for binomial determinants with consecutive or almost consecutive rows or columns, as well as calculating the generator of left nullspaces defined by some binomial matrices. In the meantime, we…

We establish a superpolynomial lower bound on the range of the permanent function on the set of $n\times n$ matrices with $\pm1$ entries.

We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

We prove two "master" convolution theorems for multivariate determinantal polynomials. The methods used include basic properties of what we call a "minor-orthogonal" ensemble as well as properties of the mixed discriminant of matrices. We…

By viewing non-commutative polynomials, that is, elements in free associative algebras, in terms of linear representations, we generalize Horner's rule to the non-commutative (multivariate) setting. We introduce the concept of Horner…

Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…

We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries…

In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…

Recently we introduced a new class of relations for Bernoulli symmetric polynomials. This manuscript shows that these relations are valid for arbitrary homogeneous symmetric polynomial. Analysis of these relations leads to the discovery of…

We tabulate polynomials in Z[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and infinity. We explain how these sets of polynomials are of…

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…

Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, etc. In this paper, we construct several new classes of permutation…

This paper aims at extending the criterion that the quasi-stability of a polynomial is equivalent to the total nonnegativity of its Hurwitz matrix. We give a complete description of functions generating doubly infinite series with totally…

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…