Related papers: Solvable Polynomials over the Gaussian Field Q(i)

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

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

In this note, we prove strong convergence of $q$-Gaussians with respect to a parameter $q$, which implies the spectrum of any self-adjoint non-commutative polynomial in $q$-Gaussians is continuously deformed with respect to $q$. With…

We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter…

For each positive integer n, we define a polynomial in the variables z_1,...,z_n with coefficients in the ring $\mathbb{Q}[q,t,r]$ of polynomial functions of three parameters q, t, r. These polynomials naturally arise in the context of…

Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how…

For a field $E$ of characteristic different from $2$ and cohomological $2$-dimension one, quadratic forms over the rational function field $E(X)$ are studied. A characterisation in terms of polynomials in $E[X]$ is obtained for having that…

We obtain some properties of the q-Narayana polynomials for q=-1 and compare them to corresponding properties for q=1.

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the…

We refine and extend quantitative bounds, on the fraction of nonnegative polynomials that are sums of squares, to the multihomogenous case.

In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.

Non-topological gauged soliton solutions called Q-balls arise in many scalar field theories that are invariant under a U(1) gauge symmetry. The related, but qualitatively distinct, Q-shell solitons have only been shown to exist for special…

We study the set of common $\mathbb{F}_q$-rational solutions of "smooth" systems of multivariate symmetric polynomials with coefficients in a finite field $\mathbb{F}_q$. We show that, under certain conditions, the set of common solutions…

A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little…

A class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations. Their zonal spherical functions are discussed in connection with $q$-orthogonal…

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large…

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$…

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…