相关论文: On real and imaginary roots of generalised Okamoto…

We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only non-negative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the…

According to the real \tau-conjecture, the number of real roots of a sum of products of sparse polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower…

We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the $\tau$ functions of rogue wave…

Starting with a rational solution to Painleve' VI, coming from a Riccati equation, using Okamoto's theory a four-parametric rational solution is obtained.

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational…

We introduce and study generalized Umemura polynomials $U_{n,m}^{(k)}(z,w;a,b)$ which are a natural generalization of the Umemura polynomials $U_n(z,w;a,b)$ related to Painlev\'e $VI$ equation. We will show that if $a=b$, or $a=0$, or $b=0$…

Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of…

We explore the structure of the p-adic automorphism group Gamma of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We initiate the study of a general dimension…

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence…

We study the problem of decomposing a non-negative polynomial as an exact sum of squares (SOS) in the case where the associated semidefinite program is feasible but not strictly feasible (for example if the polynomial has real zeros).…

We present a consistent truncation, allowing us to obtain the first degree birational transformation found by Okamoto for the sixth Painlev\'e equation. The discrete equation arising from its contiguity relation is then just the sum of six…

Let $S=\{p_1,\dots,p_s\}$ be a finite non-empty set of distinct prime numbers, let $f\in \mathbb{Z}[X]$ be a polynomial of degree $n\ge 1$, and let $S'\subseteq S$ be the subset of all $p\in S$ such that $f$ has a root in $\mathbb{Z}_p$.…

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient…

We show that there is a query expressible in first-order logic over the reals that returns, on any given semi-algebraic set A, for every point a radius around which A is conical. We obtain this result by combining famous results from…

We prove existence of infinitely many classical periodic solutions with periodic boundary conditions for a class of monotone semilinear wave equations. Our argument relies on some new estimates for the linear problem with periodic boundary…

Rational solutions of the fourth order analogue to the Painlev'e equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients…

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1]…

We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real…

We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada…

We prove the existence of infinitely many classical periodic solutions for a class of semilinear wave equations with periodic boundary conditions. Our argument relies on some new estimates for the linear problem with periodic boundary…