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Virtualconference

Authors : Dartyge, Cécile (Author of the conference)
CIRM (Publisher )

00:00

00:00

Abstract : Let

q = p^{r}

, where

p

is a prime number and

ß = (β_{1}, \dots, β_{r})

be a basis of

F_{q}

over

F_{p}

.
Any

ξ \in F_{q}

has a unique representation

ξ = \sum_{i = 1}^{r} x_{i} β_{i}

with

x_{1}, \dots, x_{r} \in F_{p}

.
The coefficients

x_{1}, \dots, x_{r}

are called the digits of

ξ

with respect to the basis

ß

.
The analog of the Rudin-Shapiro function is

R (ξ) = x_{1} x_{2} + \dots + x_{r - 1} x_{r}

. For

f \in F_{q} [X]

, non constant and

c \in F_{p}

, we obtain some formulas for the number of solutions in

F_{q}

of

R (f (ξ)) = c

. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in

F_{p}

with several variables.

This is a joint work with László Mérai and Arne Winterhof.

Keywords : finite fields; digit sums; Hooley-Katz theorem; polynomial equations; Rudin-Shapiro function

MSC Codes :
11A63 - Radix representation; digital problems
11T23 - Exponential sums
11T30 - Structure theory

Information on the Video

Film maker :

Hennenfent, Guillaume

Language :

Available date :

Conference Date :

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arXiv category :

Mathematical Area(s) :

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Download :

https://videos.cirm-math.fr/2020-11-27_Dartyge.mp4

Information on the Event

Event Title : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
Event Organizers : Rivat, Joël ; Tichy, Robert
Dates : 21/11/2020 - 27/11/2020
Event Year : 2020
Event URL : https://www.chairejeanmorlet.com/2256.html

Citation Data

DOI : 10.24350/CIRM.V.19686303
Cite this video as: Dartyge, Cécile (2020). The Rudin-Shapiro function in finite fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19686303
URI : http://dx.doi.org/10.24350/CIRM.V.19686303

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Bibliography

DARTYGE, Cécile, MÉRAI, László, et WINTERHOF, Arne. On the distribution of the Rudin-Shapiro function for finite fields. arXiv preprint arXiv:2006.02791, 2020. - https://arxiv.org/abs/2006.02791

HOOLEY, Christopher. On the number of points on a complete intersection over a finite field. Journal of number theory, 1991, vol. 38, no 3, p. 338-358. - https://doi.org/10.1016/0022-314X(91)90023-5

MULLEN, Gary L. et PANARIO, Daniel. Handbook of finite fields. CRC Press, 2013. - http://preprint.impa.br/FullText/Garcia__Thu_Mar__1_12_27_51_BRST_2012/HFFfinal.pdf

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