scholarly journals A lower bound for the number of solutions of equations over finite fields

1974 ◽  
Vol 6 (6) ◽  
pp. 448-480 ◽  
Author(s):  
Wolfgang M. Schmidt
Keyword(s):  
Lower Bound ◽  
Finite Fields ◽  
Number Of Solutions ◽  
Solutions Of Equations

scholarly journals Solutions of equations over finite fields: Enumeration via bijections

2016 ◽  
Vol 15 (07) ◽  
pp. 1650136 ◽  
Author(s):  
Ioulia N. Baoulina
Keyword(s):  
Recent Result ◽  
Finite Fields ◽  
Simple Proof ◽  
Character Sums ◽  
Alternative Proof ◽  
Number Of Solutions ◽  
Combinatorial Argument ◽  
Solutions Of Equations

We present a simple proof of the well-known fact concerning the number of solutions of diagonal equations over finite fields. In a similar manner, we give an alternative proof of the recent result on generalizations of Carlitz equations. In both cases, the use of character sums is avoided by using an elementary combinatorial argument.

scholarly journals A note on the lower bound of representation functions

2021 ◽  
pp. 1-8
Author(s):  
Xing-Wang Jiang ◽  
Csaba Sándor ◽  
Quan-Hui Yang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the Thue–Morse sequence and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. Previously, the first author proved that if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.

On the values of representation functions III

2019 ◽  
Vol 16 (03) ◽  
pp. 511-522
Author(s):  
Xing-Wang Jiang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Binary Representation ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the set of all nonnegative integers [Formula: see text] with an even number of ones in the binary representation of [Formula: see text] and let [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. In 2011, Chen proved that, if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we improve the lower bound by proving that [Formula: see text] for all [Formula: see text].

scholarly journals The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Peter Somora
Keyword(s):  
Lower Bound ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Root Functions ◽  
Number Of Zeros ◽  
Variational Solutions ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractWe consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

On the Number of Zeros Over a Finite Field of Certain Symmetric Polynomials

1980 ◽  
Vol 23 (3) ◽  
pp. 327-332
Author(s):  
P. V. Ceccherini ◽  
J. W. P. Hirschfeld
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Polynomial Equations ◽  
Symmetric Polynomials ◽  
Number Of Solutions ◽  
Number Of Zeros

A variety of applications depend on the number of solutions of polynomial equations over finite fields. Here the usual situation is reversed and we show how to use geometrical methods to estimate the number of solutions of a non-homogeneous symmetric equation in three variables.

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