Related papers: Word maps and Waring type problems

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given…

The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups have been studied recently, and in this paper we study them for finite quasisimple groups G. We…

Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least…

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups…

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…

The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of $j$ positive $k$-th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for…

In this paper, we study a density version of the Waring-Goldbach problem. Suppose that A is a subset of the primes, and the lower density of A in the primes is larger than 1-1/2k. We prove that every sufficiently large natural number n…

The Waring problem of forms concerns the expression of homogeneous multivariate polynomials as sums of powers of linear forms. This paper focuses on complex binary forms, and we solve the Waring problem for them using basic tools in algebra…

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…

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…

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…

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…

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 $\theta_2=\frac{19}{24}$,…

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…

When k > 1 and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.

Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Waring problem in numbers of class $\mathbb{N}_0$ is solved.

Let $f\in \mathbb{Q}(x)$ be a non-constant rational function. We consider "Waring's Problem for $f(x)$," i.e., whether every element of $\bbq$ can be written as a bounded sum of elements of $\{f(a)\mid a\in \mathbb{Q}\}$. For rational…

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…

Let G be a finite quasisimple group of Lie type. We show that there are regular semisimple elements x,y in G, x of prime order, and |y| is divisible by at most two primes, such that the product of the conjugacy classes of x and y contain…

The Matrix Waring problem is if we can write every matrix as a sum of $k$-th powers. Here, we look at the same problem for triangular matrix algebra $T_n(\mathbb{F}_q)$ consisting of upper triangular matrices over a finite field. We prove…