Related papers: Waring's Problem for Polynomial Rings and the Digi…

In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an…

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…

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$,…

Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with…

The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial $p$ of degree $d$ as a finite sum of $d$-{th} powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any…

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…

In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \in C[x_0,x_1,...,x_n] of degree divisible by k\ge 2 can be represented as a sum of at most…

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…

Waring's classical problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for nonabelian groups, and simple groups in particular. Here the k-th…

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…

Motivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of k-th powers of forms of degree…

We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of variables…

We investigate the existence of representations of every large positive integer as a sum of $k$-th powers of integers represented as certain diagonal forms. In particular, we consider a family of diagonal forms and discuss the problem of…

Let $G(k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$-th powers. Then for all $k\in \mathbb N$, one has \[ G(k)\le \lceil k(\log…

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers.…

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…

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…