Related papers: Hyperdeterminants of Polynomials
We discuss an integrable partial differential equation arising from the hyperdeterminant.
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…
Necessary and sufficient conditions are obtained under which the numerator of the partial derivative of a rational function holomorphic in open upper poly-halfplane is the sum of squares of polynomials.
We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions.…
In this paper we propose a conseptual framework for the observed properties of discriminants of polylinear forms. The connection with classical problems of linear algebra is shown. A new class of algebraic varieties (hypergrassmanians) is…
Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.
The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.
We present formulas for the homogenous multivariate resultant as a quotient of two determinants. They extend classical Macaulay formulas, and involve matrices of considerably smaller size, whose non zero entries include coefficients of the…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this…
Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…
In this paper, we provide the degree distribution of irreducible factors of the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial. We…
Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…
Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose…
To each associative unitary finite-dimensional algebra over a normal base, we associative a canonical multiplicative function called its determinant. We give various properties of this construction, as well as applications to the topology…
We compute the graded rank of the cohomology of the hyperplane complement associated with a quaternionic reflection group, and observe that it factors into irreducible factors with positive integer coefficients. For an irreducible group,…
An almost-toric hypersurface is parameterized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit equation.
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a…
Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their…