Yeşim Demiroğlu Karabulut
In this paper we obtain sharp results for Waring's problem over general finite rings, by using a combination of Artin-Wedderburn theory and Hensel's lemma and building on new proofs of analogous results over finite fields that are achieved using spectral graph theory. We also prove an analogue of Sárközy's theorem for finite fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Suppose n >= 2. We show that there is no integer v >= 1 such that for all commutative rings R with identity, every element of the subring J(2^n,R) of R generated by 2^n-th powers can be written in the form \pm f_1^{2^n} \pm \cdots \pm f_v^{2^n} for some f_1,...,f_v \in R and some choice of signs.
Waring Numbers of Ramified $p$-adic Rings
本文研究分歧 p 进环 R = Z_p[√[e]{p}] 的 Waring 数，计算了多种 p、e、k 取值下的 g_R(k)，并分析了 p=2 时其随 e 和 k 的变化规律。
Waring Problem for matrices over finite local rings
本文在有限局部环上研究矩阵的Waring问题，证明在特定条件下任何矩阵可表示为两个k次幂之和，并指出这两个条件是结果成立的必要条件。
Let $F$ be an algebraically closed field of characteristic $p>0$. In this paper we develop methods to represent arbitrary elements of $F[t]$ as sums of perfect $k$-th powers for any $k\in\mathbb{N}$ relatively prime to $p$. Using these methods we establish bounds on the necessary number of $k$-th powers in terms of the sum of the digits of $k$ in its base-$p$ expansion. As one particular application we prove that for any fixed prime $p>2$ and any $ε>0$ the number of $(p^r-1)$-th powers required is $\mathcal{O}\left(r^{(2+ε)\ln(p)}\right)$ as a function of $r$.
Waring numbers over finite commutative local rings
该研究将有限交换局部环上的华林数与凯莱图直径关联，通过顶点膨胀将局部环上的华林数问题归约为有限剩余域上的华林数计算，进而利用有限域上的已知结果得到局部环上华林数的若干显式结论。
On Waring numbers of henselian rings
本文研究了henselian局部环及其全分式环的$n$次Waring数，给出了基于剩余域Waring数和$n$次水平的上下界，并在多种情形下完成精确计算，特别是求得了$n$为3、4、5时的$w_n(\mathbb{Z}_p)$和$w_n(\mathbb{Q}_p)$。
We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in $\mathbb{Z}_n$ can be written as a sum of three squares. Hensel's Theorem for $p$-adic integers plays a key role. Final…
In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We prove a discriminant criterion for every 2 by 2 matrix over an order R to be sums of cubes and fourth powers over R. We also show that if q is a prime and n greater equal 2, then every n by n matrix over the ring of integers O, of a quadratic number field is a sum of q-th powers (of matrices) over O if and only if q is coprime to the discriminant of the quadrati…
We show that the Waring's number over a finite field $\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $Γ(k,q)=Cay(\mathbb{F}_{q},R_k)$ with $R_k=\{x^k : x\in \mathbb{F}_q^*\}$. We find infinite new families of exact values of $g(k,q)$ from a characterization of graphs $Γ(k,q)$ which are also Hamming graphs previously proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair $(k,q)$ with $q$ not a prime. Finally, we find a lower bound for $g(k,p)$ with $p$ prime by using that $Γ(k,p)$ …
Let $τ(n)$ be the Ramanujan $τ$-function. We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}τ(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$
The Waring function $g(k,q)$ measures the difficulty of Waring's problem for $k$th powers in the field of $q$ elements. Its calculation seems to be difficult, and many partial results have been published, notably upper bounds for certain regions of the $k$-$q$-plane. In this paper, we compute the exact value of $g(k,q)$ for two infinite families of exponent-field pairs. In these, $k$ is large compared to $q$. We use a new method of proof that is mainly combinatorial in nature.
We obtain an asymptotic formula on the Odlyzko-Stanley enumeration problem. Let $N_m^*(k,b)$ be the number of $k$-subsets $S\subseteq F_p^*$ such that $\sum_{x\in S}x^m=b$. If $m<p^{1-δ}$, then there is a constant $ε=ε(δ)>0$ such that | N_m^*(k,b)-p^{-1}{p-1 \choose k}|\leq {p^{1-ε}+mk-m \choose k}. In addition, let $γ'(m,p)$ denote the distinct Waring's number $(\mod p)$, the smallest positive integer $k$ such that every integer is a sum of m-th powers of $k$-distinct elements $(\mod p)$. The above bound implies that there is a constant $ε(δ)>0$ such for any prime $p$ and any $m<p^{1-δ}$, if …
We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the {\em $g$-invariants} of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\mathcal O$ be the ring of integers of $K$ and $g_{\mathcal O}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\mathcal O$-linear forms mus…
Waring Problem for Matrices over Finite Fields
本文证明有限域上所有矩阵可表示为两个k次幂之和，并推广至可逆、循环等情形，给出基于稳定子的判据。
We investigate a version of Waring's Problem over quaternion rings, focusing on cubes in quaternion rings with integer coefficients. We determine the global upper and lower bounds for the number of cubes necessary to represent all such quaternions.
Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that for all $s \geq t_0$, one has the expected asymptotic formula in Waring's problem over $\mathbb{F}_q[t]$ concerning sums of $s$ $k$-th powers of polynomials in $\mathbb{F}_q[t]$. For each $k$ not divisible by $p$, we derive a minor arc bound from Vinogradov-type estimates, and obtain bounds on $\widetilde{G}_q(k)$ that are quadratic in $k$, in fact linear i…
On quadratic Waring's problem in totally real number fields
该研究改进了全实域整数环的g不变量上界，给出了基于子域的次指数界，还得到格版本G(r)的一般不等式并确定了绝大多数实二次域的G(2)值。
We give a reduction formula for the Waring number $g(k,q)$ over a finite field $\mathbb{F}_q$. By exploiting the relation between $g(k,q)$ with the diameter of the generalized Paley graph $Γ(k,q)$ and by using the characterization due to Pearce and Praeger (2019) of those $Γ(k,q)$ which are Cartesian decomposable, we obtain the reduction formula $$g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a)$$ for $p$ prime and $a,b,c$ positive integers under certain arithmetic conditions. Then, we find some arithmetic conditions to apply the formula above, which allow us to obtain many infinite …
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…
On the Waring problem with Dickson polynomials modulo a prime
本文利用 Kloosterman 和与 Gauss 和的新界限，改进了素数有限域下 Dickson 多项式 Waring 问题的现有结果。
The asymptotic in Waring's problem over function fields via a singular locus in the circle method
本文在函数域上改进了瓦灵问题的渐进行为，利用Vinogradov均值定理的主要猜想，并通过处理有限域上的指数和来获得更强的结果。
Waring Problem For Triangular Matrix Algebra
本文研究有限域上三角矩阵代数的矩阵 Waring 问题，借助 Lang-Weil 估计证明大域中每个矩阵可表为三个 $k$ 次幂之和，在 $-1$ 为 $k$ 次幂时可降为两个。
For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of $g_{\mathbb Z}(n)$ squares of integral linear forms. We show that as $n$ goes to infinity, the growth of $g_{\mathbb Z}(n)$ is at most an exponential of $\sqrt{n}$. Our result improves the best known upper bound on $g_{\mathbb Z}(n)$ which is in the order of an exponential of $n$. We also define an analogous number $g_{\mathcal O}^*(n)$ for writing hermitian forms ov…
For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest…
Denote by P(K, k) the members of the field K which are sums of kth powers of field elements, by P+(K, k) the set of members of K which are sums of kth powers of totally positive elements of K. We are interested in deciding whether or not there exist integers w(K, k) and g(K, k) such that: a in P(K, k) implies that a is the sum of at most w(K, k) kth powers; a in P+(K, k) implies that a is the sum of at most g(K, k) totally positive kth powers. We will show that if w(K, 2) is finite and provided that the kth powers are dense (in a sense described explicitly in theorem 3) in K, then w(K, k) is a…
Let $s_d(p,a) = \min \{k | a = \sum_{i=1}^{k}a_i^d, a_i\in \ff_p^*\}$ be the smallest number of d-th powers in the finite field F_p, sufficient to represent the number a in F_p^*. Then $$g_d(p) = max_{a in F_p^*} s_d(p,a)$$ gives an answer to Waring's Problem mod p. We first introduce cyclotomic integers n(k,ν), which then allow to state and solve Waring's problem mod p in terms of only the cyclotomic numbers (i,j) of order d. We generalize the reciprocal of the Gaussian period equation G(T) to a C-differentiable function I(T) in Q[[T]], which also satisfies I'(T)/I(T) in Z[[T]]. We show that …
We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. Given a word $w$ in a free Lie algebra $\mathcal{L}_{r}$, it induces a word map $φ_{w}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ for every semisimple Lie algebra $\mathfrak{g}$. Given two words $w_{1}\in\mathcal{L}_{r_{1}}$ and $w_{2}\in\mathcal{L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $φ_{w_{1}}*φ_{w_{2}}:=φ_{w_{1}}+φ_{w_{2}}:\mathfrak{g}^{r_{1}+r_{2}}\rightarrow\mathfrak{g}$. We show that …
Integers that are not the sum of positive powers
本文用计算技术改进Zenkin方法，解决5≤k≤9的广义华林问题，确定不能表示为j个正k次幂之和的整数集合，并推广到G(1,k)<j<g(1,k)的有限集。
In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.
We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\mathbb{F}_q), d \geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix…
In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such results to polynomials of degrees higher than the characteristic of the field. Such results have a wide scope of applications similar to those associated with their counterparts studying consecutive intervals over prime fields instead of affine subspaces. Here we give only two immediate consequences: to a bound on the size of the intersection of orbits of p…
Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_q$ and finite cyclic rings $\mathbb{Z}/p^r\mathbb{Z}$, in the setting of finite valuation rings.
We compute the $k$th power-sums (for all $k>0$) over an arbitrary finite unital ring $R$. This unifies and extends the work of Brawley, Carlitz, and Levine for matrix rings [Duke Math. J. 1974], with folklore results for finite fields and finite cyclic groups, and more general recent results of Grau and Oller-Marcen for commutative rings [Finite Fields Appl. 2017]. As an application, we resolve a conjecture by Fortuny Ayuso, Grau, Oller-Marcen, and Rua on zeta values for matrix rings over finite commutative rings [Internat. J. Algebra Comput. 2017]. We further recast our main result via zeta v…
Kloosterman sums over finite Frobenius rings
该研究在有限交换Frobenius环上推广Kloosterman和，证明了若干扭曲Kloosterman和的恒等式，主要贡献是围绕矩计算建立相关理论。
In this work, we present a cogredient standard form of an orthogonal space over a finite local ring of odd characteristic.
We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.
We prove that the universal unramified deformation ring $R^{\mathrm{unr}}$ of a continuous Galois representation $\overlineρ: G_{F^{+}} \rightarrow \mathrm{GL}_n(k)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O} = W(k)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{\mathrm{univ}}$ is finite over the local deformation ring $R^{\mathrm{loc}}$.
In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over finite chain rings using lifted codes.
We prove the following result related to the inverse problem for universal deformation rings of group representations: Given a finite field k, denote by W(k) the ring of Witt vectors over k and by K the field of fractions of W(k). If a complete noetherian local ring R is a universal deformation ring of a representation of a finite group over k, then R \otimes_{W(k)} K is a finite étale K-algebra. As a consequence, we obtain explicit examples of characteristic zero complete noetherian local rings that can not be obtained as universal deformation rings of finite group representations. This contr…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。