Let I = 〈g1, . . . , gn〉 be a zero-dimensional ideal of R[x1, . . . , xn] such that its associated set G of polynomial equations gi(x) = 0 for all i = 1, . . . , n is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in √ I. We also provide a necessary and sufficient (numerical) condition for all the zeros of G to be in a given se...

Exceptional orthogonal polynomials were introduced by Gomez-Ullate, Kamran and Milson as polynomial eigenfunctions of second order differential equations with the remarkable property that some degrees are missing, i.e., there is not a polynomial for every degree. However, they do constitute a complete orthogonal system with respect to a weight function that is typically a rational modification ...

Rankin and Swinnerton-Dyer [R, S-D] prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A := {eiθ : π 2 ≤ θ ≤ 2π 3 }. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique “gap function” in...

We study the imaginary quadratic fields such that the Iwasawa λ2-invariant equals 1, obtaining information on zeros of 2-adic L-functions and relating this to congruences for fundamental units and class numbers. This paper explores the interplay between zeros of 2-adic L-functions and congruences for fundamental units and class numbers of quadratic fields. An underlying motivation was to study ...

For an integer n≥ 2, let p(z)=nk=1(z−αk) and q(z)= ∏n k=1(z−βk), where αk,βk are real. We find the number of connected components of the real algebraic curve {(x,y)∈R2 : |p(x+iy)|−|q(x+iy)| = 0} for some αk and βk. Moreover, in these cases, we show that each connected component contains zeros of p(z)+q(z), and we investigate the locus of zeros of p(z)+q(z). 2000 Mathematics Subject Classificati...