scholarly journals A density version of Waring’s problem

2021 ◽  
Author(s):  
Juho Salmensuu
Keyword(s):  
Waring’S Problem ◽  
Waring's Problem

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

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.

scholarly journals On Waring’s problem: Three cubes and a sixth power

2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Exponential Sums ◽  
Large Integer ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Smooth Numbers ◽  
Waring's Problem ◽  
Order Of Magnitude ◽  
Almost All ◽  
Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

An elementary estimate of G(n) in Waring's problem

1978 ◽  
Vol 24 (1) ◽  
pp. 507-513 ◽  
Author(s):  
B. M. Bredikhin ◽  
T. I. Grishina
Keyword(s):  
Waring’S Problem ◽  
Waring's Problem ◽  
Elementary Estimate

Waring's problem for polynomials of small degree

1985 ◽  
Vol 38 (3) ◽  
pp. 703-710
Author(s):  
V. P. Voloshin
Keyword(s):  
Small Degree ◽  
Waring’S Problem ◽  
Waring's Problem

scholarly journals A light-weight version of Waring's problem

2004 ◽  
Vol 76 (3) ◽  
pp. 303-316 ◽  
Author(s):  
Trevor D. Wooley
Keyword(s):  
Asymptotic Formula ◽  
Large Integer ◽  
Light Weight ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Large Solutions ◽  
Homogeneous Weight

AbstractAn asymptotic formula is established for the number of representations of a large integer as the sum of kth powers of natural numbers, in which each representation is counted with a homogeneous weight that de-emphasises the large solutions. Such an asymptotic formula necessarily fails when this weight is excessively light.

scholarly journals An Invariant Regarding Waring’s Problem for Cubic Polynomials

2009 ◽  
Vol 193 ◽  
pp. 95-110 ◽  
Author(s):  
Giorgio Ottaviani
Keyword(s):  
Secant Variety ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Veronese Variety ◽  
Waring's Problem ◽  
Cubic Polynomial ◽  
Cubic Polynomials

AbstractWe compute the equation of the 7-secant variety to the Veronese variety (P4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian.

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