ANALYTIC METHODS FOR THE DISTRIBUTION OF RATIONAL POINTS ON ALGEBRAIC VARIETIES

2007 ◽  
pp. 139-168
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Rational Points ◽  
Algebraic Varieties ◽  
Analytic Methods

scholarly journals Counting in hyperbolic spikes: The diophantine analysis of multihomogeneous diagonal equations

2018 ◽  
Vol 2018 (737) ◽  
pp. 255-300
Author(s):  
Valentin Blomer ◽  
Jörg Brüdern
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Arithmetic Functions ◽  
New Method ◽  
Algebraic Varieties ◽  
Asymptotic Formulae ◽  
Versatile Tool

AbstractA method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.

The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

2017 ◽  
Vol 24 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Shuangnian Hu ◽  
Junyong Zhao
Keyword(s):  
Number Theory ◽  
Normal Form ◽  
Finite Fields ◽  
Rational Points ◽  
System Of Equations ◽  
Algebraic Varieties ◽  
Natural Conditions ◽  
Characteristic P ◽  
Exponent Matrix ◽  
Positive Integers

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

scholarly journals Rational points over finite fields for regular models of algebraic varieties of Hodge type ࣙ1

2012 ◽  
Vol 176 (1) ◽  
pp. 413-508 ◽  
Author(s):  
Pierre Berthelot ◽  
Hélène Esnault ◽  
Kay Rülling
Keyword(s):  
Finite Fields ◽  
Rational Points ◽  
Algebraic Varieties
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