Related papers: Polynomials with and without determinantal represe…
In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into…
The positive existential theories of the sets $M_n(\mathbb N)$ without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorithmically undecidable. In further sections some easier…
We study non-linear surjective mappings on subsets of ${\cal M}_n(F)$, which preserve the zeros of some fixed polynomials in noncommuting variables. Keywords: Matrix algebra, Multilinear polynomials, Preservers.
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials,…
Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…
Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…
We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…
The polynomials that arise as coefficients when a power series is raised to the power $x$ include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such…
Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…
In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n <= 3 over a field K of characteristic zero. We do this by extending the following result for n <= 2 by F. Dillen to n <= 3: if f is a polynomial of…
We estimate the number of primes represented by a general quadratic polynomial with discriminant $\Delta$, assuming that the corresponding real character is exceptional.
In this paper, we give some determinantal and permanental representations of generalized bivariate Fibonacci p-polynomials by using various Hessenberg matrices. The results that we obtained are important since generalized bivariate…
We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular…
In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k…
We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…
A classical inequality of Sz\'asz bounds polynomials with no zeros in the upper half plane entirely in terms of their first few coefficients. Borcea-Br\"and\'en generalized this result to several variables as a piece of their…
This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result:…