Mostrando recursos 1 - 20 de 69

  1. Rational quotients of two linear forms in roots of a polynomial

    Dubickas, Artūras
    Let $f$ and $g$ be two linear forms with non-zero rational coefficients in $k$ and $\ell$ variables, respectively. We describe all separable polynomials $P$ with the property that for any choice of (not necessarily distinct) roots $\lambda_{1},\ldots,\lambda_{k+\ell}$ of $P$ the quotient between $f(\lambda_{1},\ldots,\lambda_{k})$ and $g(\lambda_{k+1},\ldots,\lambda_{k+\ell}) \ne 0$ belongs to $\mathbf{Q}$. It turns out that each such polynomial has all of its roots in a quadratic extension of $\mathbf{Q}$. This is a continuation of a recent work of Luca who considered the case when $k=\ell=2$, $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$ are both $x_{1}-x_{2}$, solved it, and raised the above problem as an open...

  2. A $p$-analogue of Euler’s constant and congruence zeta functions

    Kurokawa, Nobushige; Taguchi, Yuichiro
    A $p$-analogue of a formula of Euler on the Euler constant is given, and it is interpreted in terms of the absolute zeta functions of tori.

  3. SVV algebras

    Walker, Ruari Donald
    In 2010 Shan, Varagnolo and Vasserot introduced a family of graded algebras in order to prove a conjecture of Kashiwara and Miemietz which stated that the finite-dimensional representations of affine Hecke algebras of type $D$ categorify a module over a certain quantum group. We study these algebras, and in various cases, show how they relate to Varagnolo-Vasserot algebras and to quiver Hecke algebras which in turn allows us to deduce various homological properties.

  4. The local zeta integrals for $\mathit{GL}(2,\mathbf{C})\times \mathit{GL}(2,\mathbf{C})$

    Miyazaki, Tadashi
    In this article, for irreducible admissible infinite-dimensional representations $\Pi$ and $\Pi'$ of $\mathit{GL}(2,\mathbf{C})$, we show that the local $L$-factor $L(s,\Pi \times \Pi')$ can be expressed as some local zeta integral for $\mathit{GL}(2,\mathbf{C})\times \mathit{GL}(2,\mathbf{C})$.

  5. Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication

    Sasano, Nagatoshi
    The aim of this paper is to study relations between regular reductive prehomogeneous vector spaces (PVs) with one-dimensional scalar multiplication and the structure of graded Lie algebras. We will show that the regularity of such PVs is described by an $\mathfrak{sl}_{2}$-triplet of a graded Lie algebra.

  6. Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series

    Saad Eddin, Sumaia
    The Laurent-Stieltjes constants $\gamma_{n}(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non-principal, $(-1)^{n}\gamma_{n}(\chi)$ is simply the value of the $n$-th derivative of $L(s,\chi)$ at $s=1$. In this paper, we give an approximation of the Dirichlet $L$-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $\zeta(s)$ has no zeros in the region $|s-1|\leq 2.2093$, with $0\leq \Re{(s)}\leq 1$. This work is a continuation of [24].

  7. Some remarks on log surfaces

    Liu, Haidong
    Fujino and Tanaka established the minimal model theory for $\mathbf{Q}$-factorial log surfaces in characteristic 0 and $p$, respectively. We prove that every intermediate surface has only log terminal singularities if we run the minimal model program starting with a pair consisting of a smooth surface and a boundary $\mathbf{R}$-divisor. We further show that such a property does not hold if the initial surface is singular.

  8. A note on transcendental entire functions mapping uncountable many Liouville numbers into the set of Liouville numbers

    Lelis, Jean; Marques, Diego; Ramirez, Josimar
    In 1906, Maillet proved that given a non-constant rational function $f$, with rational coefficients, if $\xi$ is a Liouville number, then so is $f(\xi)$. Motivated by this fact, in 1984, Mahler raised the question about the existence of transcendental entire functions with this property. In this work, we define an uncountable subset of Liouville numbers for which there exists a transcendental entire function taking this set into the set of the Liouville numbers.

  9. An explicit formula of the unramified Shintani functions for $(\mathbf{GSp}_{4},\mathbf{GL}_{2} \times_{\mathbf{GL}_{1}} \mathbf{GL}_{2})$ and its application

    Gejima, Kohta
    Let $F$ be a non-archimedean local field of arbitrary characteristic. In this paper, we announce an explicit formula of the unramified Shintani functions for $(\mathbf{GSp}_{4}(F),(\mathbf{GL}_{2} \times_{\mathbf{GL}_{1}} \mathbf{GL}_{2})(F))$. As an application, we compute a local zeta integral, which represents the spin $L$-factor of $\mathbf{GSp}_{4}$.

  10. Semi-discrete finite difference schemes for the nonlinear Cauchy problems of the normal form

    Higashimori, Nobuyuki; Fujiwara, Hiroshi
    We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a proof for its convergence. The result implies that there are cases of convergence of finite difference schemes applied to ill-posed Cauchy problems.

  11. Examples of genus two fibrations with no sections on rational surfaces

    Kitagawa, Shinya
    We construct explicit examples of genus two fibrations with no sections on rational surfaces by the double covering method. For the proof of non-existence of sections, we use the theory of the virtual Mordell-Weil groups.

  12. Symmetry breaking operators for the restriction of representations of indefinite orthogonal groups $O(p,q)$

    Kobayashi, Toshiyuki; Leontiev, Alex
    For the pair $(G, G') =(O(p+1, q+1), O(p,q+1))$, we construct and give a complete classification of intertwining operators (\textit{symmetry breaking operators}) between most degenerate spherical principal series representations of $G$ and those of the subgroup $G'$, extending the work initiated by Kobayashi and Speh [Mem. Amer. Math. Soc. 2015] in the real rank one case where $q=0$. Functional identities and residue formul{æ} of the regular symmetry breaking operators are also provided explicitly. The results contribute to Program C of branching problems suggested by the first author [Progr. Math. 2015].

  13. Symmetry breaking operators for the restriction of representations of indefinite orthogonal groups $O(p,q)$

    Kobayashi, Toshiyuki; Leontiev, Alex
    For the pair $(G, G') =(O(p+1, q+1), O(p,q+1))$, we construct and give a complete classification of intertwining operators (symmetry breaking operators) between most degenerate spherical principal series representations of $G$ and those of the subgroup $G'$, extending the work initiated by Kobayashi and Speh [Mem. Amer. Math. Soc. 2015] in the real rank one case where $q=0$. Functional identities and residue formulæ of the regular symmetry breaking operators are also provided explicitly. The results contribute to Program C of branching problems suggested by the first author [Progr. Math. 2015].

  14. An algebraic proof of determinant formulas of Grothendieck polynomials

    Matsumura, Tomoo
    We give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by Hudson–Ikeda–Matsumura–Naruse in~[6] and by Hudson–Matsumura in~[7].

  15. An algebraic proof of determinant formulas of Grothendieck polynomials

    Matsumura, Tomoo
    We give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by Hudson–Ikeda–Matsumura–Naruse in [6] and by Hudson–Matsumura in [7].

  16. A character of the Siegel modular group of level 2 from theta constants

    Xiong, Xinhua
    Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are obtained. Using our theorems, some key theorems of Igusa~[2] can be recovered.

  17. A character of the Siegel modular group of level 2 from theta constants

    Xiong, Xinhua
    Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are obtained. Using our theorems, some key theorems of Igusa [2] can be recovered.

  18. Semicomplete vector fields on non-Kähler surfaces

    Guillot, Adolfo
    We investigate semicomplete meromorphic vector fields on complex surfaces, those where the solutions of the associated ordinary differential equations have no multivaluedness. We prove that if a non-Kähler compact complex surface has such a vector field, then, up to a bimeromorphic transformation, either the vector field is holomorphic, has a first integral or preserves a fibration. This extends previous results of Rebelo and the author to the non-Kähler setting.

  19. Semicomplete vector fields on non-Kähler surfaces

    Guillot, Adolfo
    We investigate semicomplete meromorphic vector fields on complex surfaces, those where the solutions of the associated ordinary differential equations have no multivaluedness. We prove that if a non-Kähler compact complex surface has such a vector field, then, up to a bimeromorphic transformation, either the vector field is holomorphic, has a first integral or preserves a fibration. This extends previous results of Rebelo and the author to the non-Kähler setting.

  20. Corrigendum On the class groups of pure function fields

    Ichimura, Humio

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