Related papers: The root distribution of polynomials with a three-…

Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally…

A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency…

We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an…

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$…

We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of…

For each $n$, let $X_n \in \{0,\ldots,n\}$ be a random variable with mean $\mu_n$, standard deviation $\sigma_n$, and let \[ P_n(z) = \sum_{k=0}^n \mathbb{P}( X_n = k) z^k ,\] be its probability generating function. We show that if none of…

Consider a connected graph $G$, and assume that every edge fails independently with probability $q$. The {\em (all-terminal) reliability polynomial} is the probability in $q$ that the spanning connected subgraph of operational edges is…

For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n\ne 0$. In this…

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results…

We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$.…

Let $\theta$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, \theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring $\Z_K$ of integers…

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability…

The exponential generating functions of {n^(n+m)} for arbitrary integer m are expressed as rational functions of the e.g.f. of {n^(n-1)} [the tree function] and then of the e.g.f. of {n^n} [the endofunction function]. The coefficients in…

A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive…

For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under…

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<=…

We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this…

If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if $f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We extend…