Apparent Singularities of Linear Difference Equations with Polynomial Coefficients

Polynomial approximation with doubling weights having finitely many zeros and singularities

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly changing'' away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. We approximate in the weighted $L_p$ (quasi) norm $\|f\|_{p, w}$ with $0<p<\infty$, where $\|f\|_{p, w} := \left(\int_{-1}^1 |f(u)|^p w(u) d…

关于具有四个奇点的广义拉梅方程的单值性和谱几何，I：半周期

On monodromy and spectral geometry of generalized Lamé equations with four singularities, I: half periods

研究具有表观参数的广义拉梅方程的酉单值性问题，通过分解表观参数空间、定义谱多项式等方法，确定单值性数据，证明广义拉梅曲线与谱曲线同构，刻画条件稳定集，得出特定条件下单值性矩阵酉的充要条件。

Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the Pöschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of the Pöschl-Teller potential, we obtain several families of explicit and global solutions of certain second order Fuchsian differential equations with an apparent singularity of characteristic exponent -2 and -1. They form orthogonal polynomials over $x\in(-1,1)$ with weight functions of the form $(1-x)^α(1+x)^β/\{(ax+b)^4q(x)^2\}$, in which $q(x)$ is a polyno…

Calabi-Yau类型费马多项式奇点中的扭转扇区与模形式

Twisted Sectors in Calabi-Yau Type Fermat Polynomial Singularities and Automorphic Forms

本文研究Calabi-Yau类型费马多项式奇点的扭转扇区与模形式的关系，利用混合Hodge结构、Riemann-Hilbert对应和镜像对称性证明生成函数与模形式的关联。

Multifractal behavior of polynomial Fourier series

We prove non-trivial upper and lower bounds for the "Spectrum of Singularities" of Fourier Series with polynomial frequencies. The Spectrum of Singularities of a function f gives the Hausdorff dimension of the set of points with a given Hölder exponent for f.

Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle

Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2β_k}, \quad |z|=1, \quad |a_k|=1, \quad β_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for $|z|=1$ and can be extended as a holomorphic and non-vanishing function to an annulus containing the unit circle. The formulas obtained are valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials, the behavior of their leading and Verblunsky coefficients, as well as give an alternative …

三维多项式向量场中单值奇点附近的法向双曲极限环面

Normally hyperbolic limit tori near monodromic singularities in 3D polynomial vector fields

本文利用平均理论与受 Christopher-Lloyd 方法启发的递归构造，研究三维多项式向量场在单值奇点附近的法向双曲极限环面最大数，并给出改进的 Hilbert 数下界。

Quasimodular forms and modular differential equations which are not apparent at cusps: I

In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Γ=Γ_0^+(N)$ generated by $Γ_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($Γ_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups $Γ_0^…

Applications of an elementary resolution of singularities algorithm to exponential sums and congruences modulo p^n

We use the resolution of singularities algorithm of [G4] to provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a prime p when x is an integer. They are proved using p-adic analogues of the theorems of [G3] on R^n sublevel set volumes and oscillatory integrals with real phase function. The proofs of these analogues use aspects of the resolution of singularities algorithms of [G4] (but for the most part not the actual resolution of singularities theorems themselves.) Unlike…

Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

Poles of Painlevé IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities…

Landau singularities and singularities of holonomic integrals of the Ising class

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurrin…