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  • 1.
    Abathun, Addisalem
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics. Addis Ababa University, Ethiopia.
    Bøgvad, Rikard
    Stockholm University, Faculty of Science, Department of Mathematics.
    Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomialsd2016In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 16, no 2, p. 167-185Article in journal (Refereed)
    Abstract [en]

    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.

  • 2. Abebaw, Tilahun
    et al.
    Bögvad, Rikard
    Stockholm University, Faculty of Science, Department of Mathematics.
    DECOMPOSITION FACTORS OF D-MODULES ON HYPERPLANE CONFIGURATIONS IN GENERAL POSITION2012In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, no 8, p. 2699-2711Article in journal (Refereed)
    Abstract [en]

    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).

  • 3.
    Borcea, Julius
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bögvad, Rikard
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shapiro, Boris
    Stockholm University, Faculty of Science, Department of Mathematics.
    Homogenized Spectral Problems for exactly solvable operators:Asymptotics of polynomial eigenfunctions2009In: Publications of the Research Institute for Mathematical Sciences, ISSN 0034-5318, E-ISSN 1663-4926, Vol. 45, p. 525-568Article in journal (Refereed)
    Abstract [en]

    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.

  • 4.
    Bögvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hägg, Christian
    Stockholm University, Faculty of Science, Department of Mathematics.
    A refinement for rational functions of Polya's method to construct Voronoi diagrams2017In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, no 1, p. 312-334Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Bögvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Källström, Rolf
    Geometric interplay between function subspaces and rikngs of differential operators2007In: Trans.Amer.Math.Soc., Vol. 359, no 5Article in journal (Refereed)
  • 6.
    Bögvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Meyer, Thomas
    On algorithmically checking whether a Hilbert series comes from a complete intersection2004In: Journal of symbolic computation, Vol. 38, no 6Article in journal (Refereed)
  • 7.
    Bögvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shapiro, Boris
    Stockholm University, Faculty of Science, Department of Mathematics.
    On mother body measures with algebraic Cauchy transform2016In: L'Enseignement mathématique, ISSN 0013-8584, E-ISSN 2309-4672, Vol. 62, no 1-2, p. 117-142Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    Decomposition of perverse sheaves on plane line arrangements2018In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, no 6, p. 2476-2487Article in journal (Refereed)
    Abstract [en]

    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.

  • 9.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    Length and decomposition of the cohomology of the complement to a hyperplane arrangementManuscript (preprint) (Other academic)
  • 10.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    Support of decomposition factors of direct images of line bundles on the open complement of a hyperplane configurationManuscript (preprint) (Other academic)
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