Related papers: Irreducibility of some spectral determinants

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued…

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most…

In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.

In the space of square matrices, we characterize row-generated subspaces, on which the determinant is an irreducible polynomial. As a corollary, we characterize square systems of polynomial equations with indeterminate coefficients, whose…

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that…

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

In this article we study the irreducibility of polynomials of the form $x^n+\epsilon_1 x^m+p^k\epsilon_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and…

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 present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The…

We give a lower bound for the degree of an irreducible factor of a given polynomial. This improves and generalizes the results obtained in [4, On the irreducible factors of a polynomial, Proc. Amer. Math. Soc., 148 (2020] 1429 -- 1437].

We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator,…

We study irreducible specializations, in particular when group-preserving specializations may not exist. We obtain a criterion in terms of embedding problems. We include several applications to analogs of Schinzel's hypothesis H and to the…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…

The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field.

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…

For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results…

It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In…

In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schr\"odinger equation with even…