Related papers: Aspects of stable polynomials
We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.
A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…
We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition…
This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.
We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at $z_i=1$ are {\it…
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions…
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…
The P-Eulerian polynomial counts the linear extensions of a labeled partially ordered set, P, by their number of descents. It is known that the P-Eulerian polynomials are real-rooted for various classes of posets P. The purpose of this…
We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…
We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the…
A class of polynomial dynamical systems called complex-balanced are locally stable and conjectured to be globally stable. In general, complex-balancing is not a robust property, i.e., small changes in parameter values may result in the loss…
We obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials
We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…
To study the zeros of octonionic polynomials, we generalize the well-known Enestrom-Kakeya Theorem to the case of octonions. In this paper, we first deal with octonionic polynomials with nonnegative and monotonic coefficients, and prove…
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this…
We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the…
An algorithm is presented here, for discovering Hopf-Bifurcation varieties of polynomial dynamical systems. It is based on the expression of specific polynomials, as sums of products of first degree polynomials, with parametrical…
In this paper we give a different proof of Kuz'min's result on the number of irreducible polynomials with the first two coefficients fixed. Our technique is to relate the question to the number of points on a curve, and to calculate the…
In this survey we determine an explicit set of generators of the maximal ideals in the ring $\mathbb R[x_1,\dots,x_n]$ of polynomials in $n$ variables with real coefficients and give an easy analytic proof of the Bass-Vasershtein theorem on…
For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an…