All Questions
Tagged with exponential-sums finite-fields
11 questions
4
votes
1
answer
338
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Bounds on quadratic character sums
I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too.
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
- 153
1
vote
1
answer
259
views
On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 563
0
votes
0
answers
92
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 563
2
votes
1
answer
154
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 563
5
votes
1
answer
287
views
Bound for sum of multiplicative character calculated over multivariate polynomial
Let $f \in \mathbb{F}_q[x_1, \dots, x_k]$ be a polynomial with $\deg f = n$, and let $\chi$ be a multiplicative character over $\mathbb{F}_q$.
Is there any known bound, possibly with conditions about $...
- 219
1
vote
0
answers
96
views
Polynomial composition utilizing polynomials in two different finite fields
At every $n\in\mathbb N$ (all polynomials are of degree $O(1)$) is there $g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n]$ at $k=\mathsf{poly}(n)$ and $g_2^{(n)}\in\mathbb F_2[x_1,\dots,...
- 13.9k
5
votes
1
answer
629
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How different can the bias of two polynomials be?
$\DeclareMathOperator\bias{bias}$I'm trying to figure out how to approach the following question:
Let $g$, $h$ be polynomials over $\mathbb{Z}_p$ (for prime $p$) with $n>1$ variables.
Denote by $\...
- 301
7
votes
1
answer
656
views
Upper bound for an exponential sum involving characters of a finite field
Let $q = p^n $ be a prime power, $\alpha\in\mathbb{F}_{q} $
a primitive element of the finite field $\mathbb{F}_q$ and denote by $\chi $ a non-trivial additive character of $\mathbb{F}_{q} $. Set
$\...
- 93
2
votes
0
answers
249
views
An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
- 353
5
votes
1
answer
328
views
Does a Kloosterman sum composed with a rational function exhibit square root cancellation?
Denote the classical Kloosterman and Salié sums, respectively, as $KL(a,b) = \sum_{r \in F_*} e(ar+\frac{b}{r})$ and $SL(a,b) =\sum_{r \in F_*} \chi(r) e(ar+\frac{b}{r})$, where $\chi(\cdot)$ is the ...
- 13k
4
votes
1
answer
307
views
When the Kloosterman sum is an integer?
Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root
of unity. We denote by
$\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\...
- 255