Related papers: Higher discriminants of polynomials
The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
We introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Then, among other things, by using the formula about representing one lambda-Sheffer polynomial in terms of other lambda-Sheffer…
We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting…
We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…
We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be…
We investigate existence of solitonic solutions for higher-order partial differential equations with polynomial nonlinearities. Using the Hirota method we obtain classification for higher-order integrable systems of equations.
Geometric modeling by constraints leads to large systems of algebraic equations. This paper studies bipartite graphs underlaid by systems of equations. It shows how these graphs make possible to polynomially decompose these systems into…
In this paper, the discriminant of homogeneous polynomials is studied in two particular cases: a single homogeneous polynomial and a collection of n-1 homogeneous polynomials in n variables. In these two cases, the discriminant is defined…
We consider systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial…
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…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
In this paper, we consider higher-order Bernoulli and poly-Bernoulli mixed type polynomials and we give some interesting identities of those polynomials arising from umbral calculus.
In order to verify programs or hybrid systems, one often needs to prove that certain formulas are unsatisfiable. In this paper, we consider conjunctions of polynomial inequalities over the reals. Classical algorithms for deciding these not…
We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…
We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…