English
The Rudin-Shapiro function in finite fields
Virtualconference
Auteurs : Dartyge, Cécile (Auteur de la Conférence)
CIRM (Editeur )
11A63 - Radix representation; digital problems
11T23 - Exponential sums
11T30 - Structure theory
CIRM (Editeur )
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Résumé :
Let $q=p^r$, where $p$ is a prime number and $ ß=(\beta_1 ,\ldots ,\beta_r)$ be a basis of $\mathbb{F}_q$ over $\mathbb{F}_p$.
Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in \mathbb{F}_q [X]$, non constant and $c\in\mathbb{F}_p$, we obtain some formulas for the number of solutions in $\mathbb{F}_q$ of $R(f(\xi ))=c$. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in $\mathbb{F}_p$ 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
Codes MSC : Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in \mathbb{F}_q [X]$, non constant and $c\in\mathbb{F}_p$, we obtain some formulas for the number of solutions in $\mathbb{F}_q$ of $R(f(\xi ))=c$. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in $\mathbb{F}_p$ 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
11A63 - Radix representation; digital problems
11T23 - Exponential sums
11T30 - Structure theory
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 01/12/2020 Date de captation : 27/11/2020 Sous collection : Research talks arXiv category : Number Theory Domaine : Number Theory Format : MP4 (.mp4) - HD Durée : 00:41:07 Audience : Researchers Download : https://videos.cirm-math.fr/2020-11-27_Dartyge.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoireOrganisateurs de la rencontre : Rivat, Joël ; Tichy, Robert Dates : 21/11/2020 - 27/11/2020 Année de la rencontre : 2020 URL Congrès : https://www.chairejeanmorlet.com/2256.html
Données de citation
DOI : 10.24350/CIRM.V.19686303Citer cette vidéo: 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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Bibliographie
- 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
