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.

scholarly journals Weyl sums, mean value estimates, and Waring's problem with friable numbers

2016 ◽  
Vol 176 (3) ◽  
pp. 249-299 ◽  
Author(s):  
Sary Drappeau ◽  
Xuancheng Shao
Keyword(s):  
Mean Value ◽  
Waring’S Problem ◽  
Waring's Problem

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.

scholarly journals APPROXIMATING THE MAIN CONJECTURE IN VINOGRADOV'S MEAN VALUE THEOREM

Mathematika ◽  
2016 ◽  
Vol 63 (1) ◽  
pp. 292-350 ◽  
Author(s):  
Trevor D. Wooley
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Main Conjecture

Waring’s problem and the circle method

Resonance ◽  
2004 ◽  
Vol 9 (6) ◽  
pp. 51-55 ◽  
Author(s):  
C. S. Yogananda
Keyword(s):  
Circle Method ◽  
Waring’S Problem ◽  
Waring's Problem

scholarly journals A note on the mean value theorem for special homogeneous spaces

1996 ◽  
Vol 143 ◽  
pp. 111-117 ◽  
Author(s):  
Masanori Morishita ◽  
Takao Watanabe
Keyword(s):  
Algebraic Group ◽  
Algebraic Variety ◽  
Homogeneous Spaces ◽  
Rational Numbers ◽  
Rational Points ◽  
Linear Algebraic Group ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean

Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

Small fractional parts of additive forms

1993 ◽  
Vol 345 (1676) ◽  
pp. 327-338 ◽  
Keyword(s):  
Exponential Sums ◽  
Common Factor ◽  
Mean Value ◽  
Large Sieve ◽  
Waring’S Problem ◽  
Mean Value Theorems ◽  
Integer Variables ◽  
Waring's Problem ◽  
Large Sieve Inequality ◽  
Additive Form

We show how the methods of Vaughan & Wooley, which have proved fruitful in dealing with Waring’s problem, may also be used to investigate the fractional parts of an additive form. Results are obtained which are near to best possible for forms with algebraic coefficients. New results are also obtained in the general case. Extensions are given to make several additive forms simultaneously small. The key ingredients in this work are: mean value theorems for exponential sums, the use of a small common factor for the integer variables, and the large sieve inequality.

scholarly journals An Error Compensation Method for Improving the Properties of a Digital Henon Map Based on the Generalized Mean Value Theorem of Differentiation

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1628
Author(s):  
Yashuang Deng ◽  
Yuhui Shi
Keyword(s):  
Chaotic System ◽  
Error Compensation ◽  
Chaotic Systems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
High Complexity ◽  
Digital World ◽  
Higher Dimensional ◽  
Generalized Mean

Continuous chaos may collapse in the digital world. This study proposes a method of error compensation for a two-dimensional digital system based on the generalized mean value theorem of differentiation that can restore the fundamental performance of chaotic systems. Different from other methods, the compensation sequence of our method comes from the chaotic system itself and can be applied to higher-dimensional digital chaotic systems. The experimental results show that the improved system is highly consistent with the real chaotic system, and it has excellent chaotic characteristics such as high complexity, randomness, and ergodicity.

Sign in / Sign up

Export Citation Format

Share Document