scholarly journals BOUNDS OF MULTIPLICATIVE CHARACTER SUMS WITH FERMAT QUOTIENTS OF PRIMES

2011 ◽  
Vol 83 (3) ◽  
pp. 456-462 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Character Sums ◽  
Multiplicative Character ◽  
Fermat Quotients ◽  
Fermat Quotient ◽  
Image Position ◽  
Multiplicative Character Sums

AbstractGiven a prime p, the Fermat quotient qp(u) of u with gcd (u,p)=1 is defined by the conditions We derive a new bound on multiplicative character sums with Fermat quotients qp(ℓ) at prime arguments ℓ.

scholarly journals ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS

2008 ◽  
Vol 50 (3) ◽  
pp. 523-529 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Absolute Constant ◽  
Character Sums ◽  
Multiplicative Character ◽  
Image Position ◽  
Multiplicative Character Sums ◽  
Bilinear Equations

AbstractWe consider the equation over a finite field q of q elements, with variables from arbitrary sets $\cA,\cB, \cC, \cD \subseteq \F_q$. The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that if for some absolute constant C > 0, then above equation has a solution for any λ ∈ q*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.

scholarly journals Stratification for Multiplicative Character Sums

2018 ◽  
Vol 2020 (10) ◽  
pp. 2881-2917 ◽  
Author(s):  
Junyan Xu
Keyword(s):  
Parameter Space ◽  
Rational Functions ◽  
Character Sums ◽  
Character Sum ◽  
Maximum Weight ◽  
Joint Work ◽  
Multiplicative Character ◽  
Multiplicative Characters ◽  
Sums Of Products ◽  
Multiplicative Character Sums

Abstract We prove a stratification result for certain families of n-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of r multiplicative characters evaluated at rational functions, and the families (with nr parameters) are obtained by allowing each of the r rational functions to be replaced by an “offset”, that is, a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least j⌊(r − 1)/2(n − 1)⌋ for 1 ≤ j ≤ n − 1 and ⌈nr/2⌉ for j = n. This result is used to obtain multivariate Burgess bounds in joint work with Lillian Pierce.

Bilinear character sums and sum-product problems on elliptic curves

2010 ◽  
Vol 53 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Omran Ahmadi ◽  
Igor Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Character Sums ◽  
Multiplicative Character ◽  
Additive Character ◽  
Product Problem ◽  
Multiplicative Character Sums

AbstractLet E be an ordinary elliptic curve over a finite field q of q elements. We improve a bound on bilinear additive character sums over points on E, and obtain its analogue for bilinear multiplicative character sums. We apply these bounds to some variants of the sum-product problem on E.

scholarly journals Double Character Sums over Subgroups and Intervals

2014 ◽  
Vol 90 (3) ◽  
pp. 376-390 ◽  
Author(s):  
MEI-CHU CHANG ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Multiplicative Group ◽  
Character Sums ◽  
Multiplicative Character ◽  
Nontrivial Estimate ◽  
Image Position ◽  
Primitive Roots ◽  
Double Character

AbstractWe estimate double sums $$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.

MULTIPLICATIVE CHARACTER SUMS OF A CLASS OF NONLINEAR RECURRENCE VECTOR SEQUENCES

2011 ◽  
Vol 07 (06) ◽  
pp. 1557-1571 ◽  
Author(s):  
ALINA OSTAFE ◽  
IGOR E. SHPARLINSKI ◽  
ARNE WINTERHOF
Keyword(s):  
Character Sums ◽  
Dimensional Case ◽  
Multiplicative Character ◽  
One Dimensional ◽  
Vector Sequences ◽  
The One ◽  
Multiplicative Character Sums

We estimate multiplicative character sums along the orbits of a class of nonlinear recurrence vector sequences. In the one-dimensional case, only much weaker estimates are known and our results have no one-dimensional analogs.

Sign in / Sign up

Export Citation Format

Share Document