Polynomial Zsigmondy theorems

Davenport-Hasse's Theorem for Polynomial Gauss Sums over Finite Fields

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport-Hasse's theorem for the entire polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes.

A decomposition theorem on differential polynomials of theta functions of high level

In this article, we prove a decomposition theorem on differential polynomials of theta functions of high level.

关于正整系数Laurent多项式的哥德巴赫定理

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。

Rudnick and Soundarajan's Theorem over Prime Polynomials for the Rational Function Field

In this paper, we use the methods of Andrade, Rudnick and Soundarajan to prove a Theorem about Lower bounds of moments of quadratic Dirichlet L-functions associated to monic irreducible polynomials over function fields.

Cantor polynomials and the Fueter-Polya theorem

A packing polynomial is a polynomial that maps the set $\mathbf{N}_0^2$ of lattice points with nonnegative coordinates bijectively onto $\mathbf{N}_0$. Cantor constructed two quadratic packing polynomials, and Fueter and Polya proved analytically that the Cantor polynomials are the only quadratic packing polynomials. The purpose of this paper is to present a beautiful elementary proof of Vsemirnov of the Fueter-Polya theorem. It is a century-old conjecture that t

A generalization of the subspace theorem for higher degree polynomials in subgeneral position

In this paper, we prove a generalization of the Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety over a number field. Our result improves and generalizes the previous results on Schmidt's subspace theorem for the case of higher degree polynomials.

An explicit polynomial analogue of Romanoff's theorem

Given a polynomial $g$ of positive degree over a finite field, we show that the proportion of polynomials of degree $n$, which can be written as $h+g^k$, where $h$ is an irreducible polynomial of degree $n$ and $k$ is a nonnegative integer, has order of magnitude $1/°g$.

Polynomial generalization of the regularization theorem for multiple zeta values

Ihara, Kaneko, and Zagier defined two regularizations of multiple zeta values and proved the regularization theorem that describes the relation between those regularizations. We show that the regularization theorem can be generalized to polynomials whose coefficients are regularizations of multiple zeta values and that specialize to symmetric multiple zeta values defined by Kaneko and Zagier.

Dirichlet's Theorem for polynomial rings

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many polynomials c(X)\in F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.

Polynomials of Binomial Type and Lucas' Theorem

We present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open question whether we then have accounted for all sequences in finite characteristic which satisfy the Binomial Theorem.

Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemerédi theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and $v_1, ..., v_t$ are nonzero vectors in $\mathbb{F}_p^D$, we show that each subset of $\mathbb{F}_p^D$ lacking a nontrivial configuration of the form $$ x, x + v_1 P_1(y), ..., x + v_t P_t(y)$$ has at most $O(p^{D-c})$ elements. In doing so, we apply the notion of Gowers norms along a vector adapted from ergodic theory, which extends the classical concept of Gow…

A note on the Brawley-Carlitz theorem on irreducibility of composed products of polynomials over finite fields

We give a new proof of the Brawley-Carlitz theorem on irreducibility of the composed products of irreducible polynomials. Our proof shows that associativity of the binary operation for the composed product is not necessary. We then investigate binary operations defined by polynomial functions, and give a sufficient condition in terms of degrees for the requirement in the Brawley-Carlitz theorem.

An Analogue of Weil's Converse Theorem for Harmonic Maass Forms of Polynomial Growth

We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.

辛钦定理在具有多项式衰减条件的非齐次同时逼近中的应用

Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

本文研究非齐次同时逼近中多项式衰减条件下的辛钦定理，解决(n,m)=(1,2)情况下的单调性条件去除问题。

Quantitative subspace theorem and general form of second main theorem for higher degree polynomials

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem…

A consequence of the factorisation theorem for polynomial orbits on nilmanifolds

We discuss a consequence of Green and Tao's factorisation theorem for polynomial orbits on nilmanifolds, adjusted to the requirements of certain arithmetic applications. More precisely, we prove a generalisation of Theorem 16.4, Acta Arith. 154 (2012), 235-306, by slightly rearranging its proof. The thus achieved strengthening of the result removes an oversight in the above-cited paper which resulted from the previously too weak conclusion. Since this type of result proved essential for further applications, we take the opportunity to discuss it in more detail.

A short proof of the canonical polynomial van der Waerden theorem

We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by Girao [arXiv:2004.07766].

A Polynomial Identity Implying Schur's Partition Theorem

We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and $q$-series identities.

A Polynomial Variation on Meinardus' Theorem

We develop a polynomial analogue of Meinardus' Thoerem for bivariate Euler products and apply it to the study of complex multiplicatively weighted partitions.

Two addition theorems on polynomials of prime variables

We extend a recent result of Khalfalah and Szemeredi to the polynomials of prime variables.

Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case

We obtain quantitative bounds in the polynomial Szemerédi theorem of Bergelson and Leibman, provided the polynomials are homogeneous and of the same degree. Such configurations include arithmetic progressions with common difference equal to a perfect kth power.

The polynomial analogue of a theorem of Renyi

Renyi's result on the density of integers whose prime factorizations have excess multiplicity has an analogue for polynomials over a finite field.

On the Almkvist-Meurman theorem for Bernoulli polynomials

Almkvist and Meurman showed that if h and k are integers, then so is $k^n(B_n(h/k) - B_n)$ where $B_n(u)$ is the Bernoulli polynomial. We give here a new and simpler proof of the Almkvist-Meurman theorem using generating functions. We describe some properties of these numbers and prove a common generalization of the Almkvist-Meurman theorem and a result of Gy on Bernoulli-Stirling numbers. We then give a simple generating function proof of an analogue of the Almkvist-Meurman theorem for Euler polynomials, due to Fox.

Polynomial Identities Implying Capparelli's Partition Theorems

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate triple sums and products. A couple of new Slater type identities are also noted.

New infinite hierarchies of polynomial identities related to the Capparelli partition theorems

We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the $q\mapsto 1/q$ duality transformation of the base identities and some related partition theoretic relations.

Polynomial Schur's theorem

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. For polynomials that are not $2$-Ramsey, we characterise all $2$-colourings of $\mathbb{N}$ that are not $2$-Ramsey, revealing that certain divisibility barrier is the only obstruction to $2$-Ramseyness for $x+y=p(z)$.

On the polynomial Szemerédi theorem in finite fields

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $γ>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-γ}$ contains a nontrivial polynomial progression $x,x+P_1(y),\dots,x+P_m(y)$, provided the characteristic of $\mathbb{F}_q$ is large enough.

Further bounds in the polynomial Szemerédi theorem over finite fields

We provide upper bounds for the size of subsets of finite fields lacking the polynomial progression $$ x, x+y, ..., x+(m-1)y, x+y^m, ..., x+y^{m+k-1}.$$ These are the first known upper bounds in the polynomial Szemerédi theorem for the case when polynomials are neither linearly independent nor homogeneous of the same degree. We moreover improve known bounds for subsets of finite fields lacking arithmetic progressions with a difference coming from the set of $k$-th power residues, i.e. configurations of the form $$ x, x+y^k, ..., x+(m-1)y^k.$$ Both results follow from an estimate of the number …

Improved Bounds on Sarkozy's Theorem for Quadratic Polynomials

We extend the best known bound on the largest subset of {1,2,...,N} with no square differences to the largest possible class of quadratic polynomials.

A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic

We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.

两两独立联合相交多项式的多维Szemerédi定理

Multidimensional Szemerédi Theorem for Pairwise Independent Jointly Intersective Polynomials

该研究针对可交换变换的联合相交性猜想，通过开发规避多项式van der Waerden定理的新技术，证明了两两独立多项式情形下的猜想，获得了超越经典单变换的任意长度序列的首个一般性结果。