Alex Eskin, Hee Oh
We study a refined version of the Linnik problem on the asymptotic behavior of the number of representations of integer $m$ by an integral polynomial as $m$ tends to infinity. We assume that the polynomial arises from invariant theory, and use methods from the theory of unipotent flows.
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This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,μ_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$μ_{N}$'s involve sequences of positive integers ; in this IV-1, by consid…
Polynomially effective equidistribution for certain unipotent subgroups in quotients of perfect Lie groups
本文证明了完美李群商中某些单位元子群轨道的有效等分布定理，并应用该结果得到不定二次型的有效Oppenheim猜想版本。
Effective equidistribution for some one parameter unipotent flows
本文针对 SL₂(ℂ) 及 SL₂(ℝ)×SL₂(ℝ) 的算术商空间，利用 Margulis 函数、关联几何工具和谱间隙，证明了 SL₂(ℝ) 幂幺子群轨道的有效均匀分布定理，并给出多项式误差率。
On the enumeration of orbits of unipotent groups over finite fields
本文证明有限域上 $\mathbf{Z}$-定义幂零群的线性轨道与共轭类枚举是“野”的，即任意概形模 $q^n$ 的点数可一致表示为这些群轨道数与洛朗多项式的组合。
In this paper, we formulate an analogue of Waring's problem for an algebraic group $G$. At the field level we consider a morphism of varieties $f\colon \mathbb{A}^1\to G$ and ask whether every element of $G(K)$ is the product of a bounded number of elements $f(\mathbb{A}^1(K)) = f(K)$. We give an affirmative answer when $G$ is unipotent and $K$ is a characteristic zero field which is not formally real. The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of $G(\mathcal{O})$ can be written as a produc…
We introduce three simple polynomial maps with integer coefficients that have interesting dynamical properties modulo primes.
In this paper are characterized the polynomials, in terms of their coefficients, that have all their orbits dense in the set of 3-adic integers.
We obtain effective bounds on the heights of algebraic integers whose orbits contain multiplicatively dependent values modulo S-integers. Our method is based on a new upper bound on the so-called S-height of polynomial values over the ring of integers of $\mathbb{K}$. Our results provide an effective variant of a recent result of A.Bérczes, A.Ostafe, I.E.Shparlinski and J.H.Silverman (arXiv:1811.04971) on multiplicative dependence modulo a finitely generated subgroup by eliminating the use of non-effective results by K.F.Roth and G.Faltings.
We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the $n$-diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion which allows one to the give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than $4$.
We give a simpler proof of an earlier result giving an asymptotic estimate for the number of integral matrices, in large balls, with a given monic integral irreducible polynomial as their common characteristic polynomial. The proof uses equidistributions of polynomial trajectories on SL(n,R)/SL(n,Z), which is a generalization of Ratner's theorem on equidistributions of unipotent trajectories. We also compute the exact constants appearing in the above mentioned asymptotic estimate.
Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori
本文通过研究三次二次型在整数上的取值，验证了伯里-塔布猜想在三维扭体中的适用性，解决了特殊对角型在不同缩放下的分布问题。
We solve a Lehmer-type question about the Mahler measure of integer-valued polynomials.
We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.
Expressing an integer as a sum of cubes of polynomials
本文证明无穷多整数可表示为四个整系数多项式立方和，且每个整数可表示为五个多项式立方和，并给出1和2的具体恒等式。
Three integers arising from Bézout's identity and resultants of integer polynomials
本文研究了由贝祖恒等式、两个互质整数多项式的余式及简化余式自然产生的三个整数，建立了新的整除关系，并提出两个猜想。
We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k \geq 2$ is and integer and $g \colon \mathbb{N} \to \mathbb{R}$ is a generalised polynomial such that $g(k^n) = 0$ for all $n \geq 0$ then there exists infinitely many $m \in \mathbb{N}$, not divisible by $k$, such that $g(mk^n) = 0$ for some $n \geq 0$. As a consequence, we obtain a complete characterisation of sequences which are simultaneously automatic and generalised polynomial.
A Goldbach theorem for Laurent polynomials with positive integer coefficients
本文证明了正整系数Laurent多项式情形下的哥德巴赫猜想类比定理。
Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemerédi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{P}-1$ (or similarly of $\mathbb{P}+1$). Using the Furstenberg Correspondence Principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\…
The following problem was posed by user "Kevin" on Mathoverflow. How to prove this polynomial always has integer values at all integers? $$P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m} \binom{x+j}{j} \binom{x-1}{j} \binom{j}{i} \binom{m}{i} \binom{i}{m-j} \frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$ We provide an answer.
We obtain a close to the best possible version of the large sieve inequality with amplitudes given by the values of a polynomial with integer coefficients of degree $\geq 2$.
In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo power…
In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer $n$ and any set of $n$ distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree $n$, height at most $H$ and Galois group $S_n$ whose roots are close to the given $n$ points.
Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. Taking quadratic integer rings as our base, we obtain as characteristic polynomials some low-degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix.
We will study p-adic invariant integerals involving trigonometric functions
A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree and bounded height. Moreover, we obtain asymptotic formulas for the number of decomposable monic polynomials of even degree. For example, the number of monic sextic integer polynomials which are decomposable and of height at most $H$ is asymptotic to $(16ζ(3)-5/4)H^3$ as $H \to \infty$.
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number
We construct parametric families of (monic) reducible polynomials having two roots very close to each other.
Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $α(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\ (mod\ n)$ is solvable. We describe a method that allows to determine the function $α$ associated to polynomials of the form $c_1x_1^k+c_2x_2^k+\cdots+c_tx_t^k$. Then we apply this method to polynomials that involve sums and differences of squares, mainly to the polynomials $x^2+y^2, x^2-y^2$ and $x^2+y^2+z^2$.
In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height $H$ among all quadratic integer polynomials…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
A correction to a result of Chinburg and Henriksen on powers of integer polynomials
本文修正了Chinburg和Henriksen于1976年提出的关于整数多项式k次幂线性组合中最小正整数m(k)的公式错误。
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.
This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+ε})$. Strassen's well known factoring approach is a special case of our method. We will also consider a possibility to improve upon the complexity bound.
We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).
On the greatest common divisor of integer parts of polynomials
本文受Bergelson和Richter的问题启发，研究了正整数与多项式整数部分构成互素元组的计数问题，并给出了误差项强度依赖于首项系数丢番图性质的渐近公式。
In this paper, we exploit the r-Stirling numbers of both kinds in order to give explicit formulae for the values of the high order Bernoulli numbers and polynomials of both kinds at integers. We give also some identities linked the r-Stirling numbers and binomial coefficients.
A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting
本文在Ekedahl导出的ℓ-adic设定中，证明了关于unipotent线性代数群的紧开子群的上同调定理，结合了Pink公式和自动形式 perverse sheaves的应用。
In this paper we construct two types of Hessenberg matrices with the properties that every weighted isobaric polynomial (WIP) appears as a determinant of one of them, and as the permanent of the other. Every integer sequence which is linearly recurrent is representable by (an evaluation of) some linearly recurrent sequence of WIPs. WIPs are symmetric polynomials written on the elementary symmetric polynomial basis. Among them are the generalized Fibonacci polynomials and the generalized Lucas polynomials, which already have these sweeping representing properties. Among the integer sequences di…
Let $F(\boldsymbol x)$ be a homogeneous polynomial in $n \ge 1$ variables of degree $1 \leq d \leq n$ with integer coefficients so that its degree in every variable is equal to $1$. We give some sufficient conditions on $F$ to ensure that for every integer $b$ there exists an integer vector $\boldsymbol a$ such that $F(\boldsymbol a) = b$. The conditions provided also guarantee that the vector $\boldsymbol a$ can be found in a finite number of steps.
We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We verify several special cases of the relevant conjectures; in particular, we find necessary and sufficient conditions for a set A to tile the integers if A is a direct sum of cyclic subsets.
We study a natural action of the Heisenberg group of integer unipotent matrices of the third order on distribution space of a two-dimensional local field for a flag on a two-dimensional scheme.
Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j \in \{1,\dots,n-1\})$ define finitely many polynomials of $\mathcal{F}_n$.
We study the uniform distribution of the polynomial sequence $λ(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $λ(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational coefficient other than the constant term. In the case of even degree, we prove a stronger result: $λ(P)$ intersects every congruence class modulo every integer if and only if $P(x)$ has at least one irrational coefficient other than the constant term.
Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies of (N_0,+). This implies that [f] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V). In the process, we modify the concept of polynomial closure in such a way that every subset of D has a finite polynomially dense subset. The results generalize to Int(S,V), the ring of integer-valued polyno…
We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.
If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case) then (a) implies (b) (obviously), (b) implies (c), and neither converse holds. Condition (b) is equivalent to the set S being the set of integer values taken by some k-tuple of polynomials with rational coefficients as the variables range through the integers. We also show…
Weighted averages of Bernoulli polynomials and the Gamma function over regular integers modulo $n$
该研究推导了模n正则整数上伯努利多项式与伽马函数加权平均的显式渐近公式，并将其应用于恒等函数、默比乌斯函数等算术函数的渐近分析。
An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q({\mathbf x} + {\mathbf v})$, where $Q$ is an integral quadratic form in the variables ${\mathbf x} = (x_1, \ldots, x_n)$ and ${\mathbf v}$ is a vector in ${\mathbb Q}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c{\mathbf v} \in {\mathbb Z}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equ…
The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in some depth. In this paper, we show previously unknown connections between BSD representations and the Stern polynomial. We derive a weight-distribution theorem for $i$-bit BSD representations of an integer $n$ in terms of the coefficients and degrees of the terms of the Stern polynomial of $2^i-n$. We then show new recursions on Stern polynomials, and from the…
It is well known, that if polynomial with rational coefficients of degree $n$ takes integer values in points $0,1,...,n$ then it takes integer values in all integer points. Are there sets of $n+1$ points with the same property in other integral domains? We show that answer is negative for the ring of Gaussian integers $\mathbb{Z}[i]$ when $n$ is large enough. Also we discuss the question about minimal possible size of set, such that if polynomial takes integer values in all points of this set then it is integer-valued.
The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and Smith normal form of integral matrices with integer parameters are also given.
We give an instant evaluation of multiple Zeta function at non-positive integers by elementary methods and discuss the Fourier theory (on unit interval) of the product of Bernoulli polynomials.We also show that the polynomial expression for Hurwitz Zeta function(at non-positive integral values of the first variable) and the polynomial expression for Bernoulli polynomials are equivalent.
Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers
本文引入与正则除数系和任意正整数集相关的广义分圆多项式，证明其具有整数系数并可分解为经典分圆多项式的乘积，同时给出新的 Menon 型恒等式。
Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $δ\ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the inequality $\|f(x)\| \le δ$ with $x \in [X,2X] \cap \mathbb{N}$.
We present an algorithm to find invariant poynomial transformations of integer sequences, using the classical invariant theory approach.
Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to prove that certain rational sums involving even powers of those polynomials are integers whenever they are evaluated at integers. This confirms two conjectures of Z.-W. Sun. We also conjecture that many of these results have neat $q$-analogues.
Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring of rational integers to rings of algebraic integers that are unique factorization domains, with special interest in the ring of Gaussian integers. We apply our method to establish strong properties of nice polynomials whose degree is a prime power. We present a considerable reduction of the system of equations for nice polynomials of arbitrary degree with th…
A study of some recent irreducibility criteria for polynomials having integer coefficients
该研究针对有理数域上的整系数多项式，梳理介绍了其相关的最新不可约性检验准则，为数论领域多项式不可约性的判定研究提供了近期成果的梳理参考。
Using the following $_4F_3$ transformation formula $$ \sum_{k=0}^{n}{-x-1\choose k}^2{x\choose n-k}^2=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}^2{x+k\choose 2k}, $$ which can be proved by Zeilberger's algorithm, we confirm some special cases of a recent conjecture of Z.-W. Sun on integer-valued polynomials.
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