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.

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.

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 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.

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.

Zeros of random polynomials over finite fields

1992 ◽  
Vol 111 (2) ◽  
pp. 193-197 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Random Polynomials ◽  
Polynomials Over Finite Fields ◽  
Image Position

Let be the finite field with q elements (q a prime power), let r 1 and let X1, , Xr be independent indeterminates over . We choose an arbitrary and a d 1 and consider

Diagonal Equations Over Large Finite Fields

1984 ◽  
Vol 36 (2) ◽  
pp. 249-262 ◽  
Author(s):  
Charles Small
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Finite Extension ◽  
Extension Field ◽  
Explicit Bound ◽  
Image Position

We consider polynomials of the formwith non-zero coefficients ai in a finite field F. For any finite extension field K ⊇ F, let fk:Kn → K be the mapping defined by f. We say f is universal over K if fK is surjective, and f is isotropic over K if fK has a non-trivial “kernel“; the latter means fK(X) = 0 for some 0 ≠ x ∊ Kn.We show (Theorem 1) that f is universal over K provided |K| (the cardinality of K) is larger than a certain explicit bound given in terms of the exponents d1,…, dn. The analogous fact for isotropy is Theorem 2.It should be noted that in studying diagonal equationswe fix both the number of variables n and the exponents di, and ask how large the field must be to guarantee a solution.

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