The polynomial alpha=xy Pi(m)(i=3)(a(i)x+y)is an element of C[x,y] determines a plane central line arrangement alpha = 0. We compute explicitly multivariate Bernstein-Sato ideals of alpha by using the decomposition behavior of the D-2-module M-alpha(beta)=C[x,y,1/alpha]alpha(beta) given in earlier work by the authors. Our results are partially special cases of recent work in much greater generality, by Maisonobe on free hyperplane arrangements and Budur et al.) as well as Bath; however our proof is independent and gives some more information on different variants of Bernstein-Sato ideals in the simple plane case.

A conjecture of Khang Tran  claims that for an arbitrary pair of polynomials A(z) and B(z), every zero of every polynomial in the sequence {P_n(z)} satisfying the three-term recurrence relation of length k

P_n(z) + B(z)P_{n−1}(z) + A(z)P_{n−k}(z) = 0

with the standard initial conditions P_0(z) = 1, P_{−1}(z) = · · · = P_{−k+1}(z) = 0 which is not a zero of A(z) lies on the real (semi)-algebraic curve C  given by

Im(( B^k(z)/ A(z)) = 0 and 0 ≤ (−1)^k ≤ Re(( B^k(z)/ A(z)) ≤ k^k (k − 1)^{k−1}. In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by

P_n(z) + B(z)P_{n−l}(z) + A(z)P_{n−k}(z) = 0, with coprime k and l and the same standard initial conditions as above, every root of P_n(z) which is not a zero of A(z)B(z) belongs to the real algebraic curve C_{l,k} given by

Im(( B^k(z)/ A(z)) = 0.

Let A be a hyperplane arrangement in C-n. We prove in an elementary way that the number of decomposition factors as a perverse sheaf of the direct image Rj(*) C-(U) over tilde[n] of the constant sheaf on the complement (U) over tilde to the arrangement is given by the Poincare polynomial of the arrangement. Furthermore, we describe the decomposition factors of Rj(*) C-(U) over tilde[n] as certain local cohomology sheaves and give their multiplicity. These results are implicitly contained, with different proofs, in Looijenga [Contemp. Math., 150 (1993), pp. 205-228], Budur and Saito [Math. Ann., 347 (2010), no. 3, 545-579], Petersen [Geom. Topol., 21 (2017), no. 4, 2527-2555], and Oaku [Length and multiplicity of the local cohomology with support in a hyperplane arrangement, arXiv: 1509.01813v1].

On the complement X = C-2 - U-i=1(n) L-i to a central plane line arrangement U-i=1(n) L-i subset of C-2, a locally constant sheaf of complex vector spaces L-a is associated to any multi-index aC(n). Using the description of MacPherson and Vilonen of the category of perverse sheaves [7, 8], we obtain a criterion for the irreducibility and number of decomposition factors of the direct image j : x -> C-2 as a perverse sheaf, where j:XC2 is the canonical inclusion.

We prove that as n -> infinity, the zeros of the polynomial F-2(1) 9-n, (an + 2) (an + 1) ; z] cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999-2001) to the case of a complex parameter alpha and partially proves a conjecture made by the authors in an earlier work.

Given a complex polynomial P with zeroes z(1),..., z(d), we show that the asymptotic zero-counting measure of the iterated derivatives Q((n)), n = 1, 2,..., where Q = R/P is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with z(1),...,z(d). This refines Polya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in C-m.

We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials [GRAPHICS] , using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525-568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275-287, 2001), Driver and Duren (Algorithms 21(1-4):147-156, 1999), and Duren and Guillou (J Approx Theory 111(2):329-343, 2001), and present experimental evidence to substantiate it.

Below we discuss the existence of a mother body measure for the exterior inverse problem in potential theory in the complex plane. More exactly, we study the question of representability almost everywhere (a.e.) in C of (a branch of) an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. Firstly, we present a large class of algebraic functions for which there (conjecturally) always exists a positive measure with the above properties. This class was discovered in our earlier study of exactly solvable linear differential operators. Secondly, we investigate in detail the representability problem in the case when the Cauchy transform satisfies a quadratic equation with polynomial coefficients a.e. in C. Several conjectures and open problems are posed.

Let alpha(1), ... , alpha(m) be linear functions on C-n and X = C-n \ V(alpha), where alpha = Pi(m)(i=1) alpha(i) and V(alpha) = {p is an element of C-n : alpha(p) = 0}. The coordinate ring O-X = C[x](alpha) of X is a holonomic A(n)-module, where A(n) is the n-th Weyl algebra, and since holonomic A(n)-modules have finite length, O-X has finite length. We consider a twisted variant of this A(n)-module which is also holonomic. Define M-alpha(beta) to be the free rank 1 C[x](alpha)-module on the generator alpha(beta) (thought of as a multivalued function), where alpha(beta) = alpha(beta 1)(1) ... alpha(beta m)(m) and the multi-index beta = (beta(1), ... , beta(m)) is an element of C-m. It is straightforward to describe the decomposition factors of M-alpha(beta), when the linear functions alpha(1), ... , alpha(m) define a normal crossing hyperplane configuration, and we use this to give a sufficient criterion on beta for the irreducibility of M-alpha(beta), in terms of numerical data for a resolution of the singularities of V(alpha).

Consider a homogenized spectral pencil of exactly solvable linear differential operators T-lambda = Sigma(k)(i=0) Q(i)(z)lambda(k-i) d(i)/dz(i), where each Q(i)(z) is a polynomial of degree at most i and lambda is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values lambda(n,j), 1 <= j <= k, of the spectral parameter lambda such that the operator T-lambda has a polynomial eigenfunction p(n,j)(z) of degree n. These eigenfunctions split into k different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits Psi(j)(Z) = lim(n ->infinity) P'(n,j) (z)/lambda(n,j)p(n,j)(z) exist, are analytic and satisfy the algebraic equation Sigma(k)(i=0)Q(i)(z)Psi(i)(j)(z) = 0 almost everywhere in CP1. As a consequence we obtain a class of algebraic functions possessing a branch near infinity is an element of CP1 which is representable as the Cauchy transform of a compactly supported probability measure.