waring problem
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Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.



2020 ◽  
pp. 5-55
Author(s):  
Nikolai Vavilov ◽  

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.



Author(s):  
Enrico Carlini ◽  
Huy Tài Hà ◽  
Brian Harbourne ◽  
Adam Van Tuyl
Keyword(s):  


Author(s):  
Enrico Carlini ◽  
Huy Tài Hà ◽  
Brian Harbourne ◽  
Adam Van Tuyl
Keyword(s):  


Author(s):  
Enrico Carlini ◽  
Huy Tài Hà ◽  
Brian Harbourne ◽  
Adam Van Tuyl
Keyword(s):  


2015 ◽  
Vol 210 (1) ◽  
pp. 81-100 ◽  
Author(s):  
Chun Yin Hui ◽  
Michael Larsen ◽  
Aner Shalev


2015 ◽  
Vol 3 ◽  
Author(s):  
MICHAEL LARSEN ◽  
PHAM HUU TIEP

Let $w_{1}$ and $w_{2}$ be nontrivial words in free groups $F_{n_{1}}$ and $F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups $G$, there exist subsets $C_{1}\subseteq w_{1}(G)$ and $C_{2}\subseteq w_{2}(G)$ such that $|C_{i}|=O(|G|^{1/2}\log ^{1/2}|G|)$ and $C_{1}C_{2}=G$. In particular, if $w$ is any nontrivial word and $G$ is a sufficiently large finite nonabelian simple group, then $w(G)$ contains a thin base of order $2$. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] for the classical Waring problem. Further results concerning thin bases of $G$ of order $2$ are established for any finite group and for any compact Lie group $G$.



2015 ◽  
Vol 137 (5) ◽  
pp. 1401-1430 ◽  
Author(s):  
Robert M. Guralnick ◽  
Pham Huu Tiep


2014 ◽  
Vol 7 (2) ◽  
pp. 239-244
Author(s):  
Chris Fuller ◽  
David Prier ◽  
Karissa Vasconi
Keyword(s):  


2013 ◽  
Vol 82 (284) ◽  
pp. 2333-2341 ◽  
Author(s):  
Tianxin Cai ◽  
Deyi Chen
Keyword(s):  


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