More on the Waring Problem

2020 ◽  
pp. 127-131
Author(s):  
Enrico Carlini ◽  
Huy Tài Hà ◽  
Brian Harbourne ◽  
Adam Van Tuyl
Keyword(s):  
Waring 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 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.

On the easier Waring problem for powers of primes. II

1939 ◽  
Vol 35 (2) ◽  
pp. 149-165 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Waring Problem ◽  
Image Position ◽  
Positive Integers

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:The density of each of the sets of integersis positive.

On the Waring-Siegel Theorem

1953 ◽  
Vol 5 ◽  
pp. 439-450 ◽  
Author(s):  
R. G. Ayoub
Keyword(s):  
Waring Problem ◽  
Image Position ◽  
Siegel Theorem

The Waring problem deals with the decomposition of integers into sums of kth powers. Consider

On the easier Waring problem for powers of primes. I

1937 ◽  
Vol 33 (1) ◽  
pp. 6-12 ◽  
Author(s):  
Paul Erdös
Keyword(s):  
Famous Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Bounded Number ◽  
True Value

The famous theorem of Schnirelmann states that a constant c exists such that every integer greater than one may be expressed as the sum of at most c primes. Recently, Heilbronn, Landau and Scherk proved that this holds with c = 71. Probably the true value of c is 3. Another well-known theorem (Hilbert's solution of Waring's problem) is that every integer is the sum of a bounded number of positive kth powers. If we omit the restriction that all the kth powers are positive, the problem is referred to as the easier Waring problem and the proof of the result is then much simpler.

Algebra of the Waring Problem for Forms

2020 ◽  
pp. 121-126
Author(s):  
Enrico Carlini ◽  
Huy Tài Hà ◽  
Brian Harbourne ◽  
Adam Van Tuyl
Keyword(s):  
Waring Problem

scholarly journals The solution to the Waring problem for monomials and the sum of coprime monomials

2012 ◽  
Vol 370 ◽  
pp. 5-14 ◽  
Author(s):  
Enrico Carlini ◽  
Maria Virginia Catalisano ◽  
Anthony V. Geramita
Keyword(s):  
Waring Problem
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