Asymptotics of Zeros of Polynomials Arising from Rational Integrals

Higher Order Degenerate Hermite-Bernoulli Polynomials Arising from $p$-Adic Integrals on $mathbb{Z}_p$

Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate $p$-adic invariant integrals on $mathbb{Z}_p$. We give interesting identities and properties of these polynomials that are derived using the generating functions and $p$-adic integral equations. Several familiar and new results are shown to follow as special cases. So...

LINEAR ESTIMATE OF THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A KIND OF QUINTIC HAMILTONIANS

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

On the Zeroes of Two Families of Polynomials Arising from Certain Rational Integrals

Vortices and Polynomials

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are g...

On Classifications of Random Polynomials

&nbsp;Let $ a_0&nbsp;(omega),&nbsp;a_1&nbsp;(omega),&nbsp;a_2&nbsp;(omega), dots,&nbsp;a_n&nbsp;(omega)$&nbsp;be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr,&nbsp;A)$. There are many known results for the expected number of real zeros of a polynomial&nbsp;$ a_0&nbsp;(omega) psi_0(x)+&nbsp;a_1&nbsp;(omega)psi_1 (x)+,&nbsp;a_2&nbsp;(omega)psi_2 (x)+...