Estimation of certain exponential sums arising in complexity theory

Jean Bourgain

doi:10.1016/j.crma.2005.03.008

Number Theory

Estimation of certain exponential sums arising in complexity theory
[Estimations de certaines sommes exponentielles en theorie de complexité]

Présenté par : Jean Bourgain

Jean Bourgain ¹

¹ Institute for Advanced Study, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 627-631.

Résumé
Abstract

On démontre que la corrélation sur ${0, 1}^{n}$ de la fonction parité et un polynôme $p (x_{1}, \dots, x_{n}) \in Z [X_{1}, \dots, X_{n}] (\mod q)$ , q un entier impair donné et $p (X)$ de degré d arbitraire mais fixé, est exponentiellement petite en n pour $n \to \infty$ . On obient une application en théorie de complexité où la question trouve son origine.

It is shown that the correlation on ${0, 1}^{n}$ between parity and a polynomial $p (x_{1}, \dots, x_{n}) \in Z [X_{1}, \dots, X_{n}] (\mod q)$ , q a fixed odd number and $p (X)$ of degree d arbitrary but fixed, is exponentially small in n as $n \to \infty$ . An application to circuit complexity, from where the problem originates, is given.

Reçu le : 2005-02-03
Accepté le : 2005-03-01
Publié le : 2005-04-27

DOI : 10.1016/j.crma.2005.03.008

Affiliations des auteurs :

Jean Bourgain ¹

¹ Institute for Advanced Study, Princeton, NJ 08540, USA

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Jean Bourgain. Estimation of certain exponential sums arising in complexity theory. Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 627-631. doi : 10.1016/j.crma.2005.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.008/

Bibliographie
Cité par

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[2] J.-Y. Cai; F. Green; T. Thierauf On the correlation of symmetric functions, Math. Systems Theory, Volume 29 (1996) no. 3, pp. 245-258

[3] E. Dueñez, S. Miller, A. Roy, H. Straubing Incomplete quadratic exponential sums in several variables, preprint

[4] F. Green The correlation between parity and quadratic polynomials mod3, J. Comput. System Sci., Volume 69 (2004) no. 1, pp. 28-44

[5] A. Hajnal; W. Maass; P. Pudlák; M. Szegedy; G. Turán Threshold circuits of bounded depth, J. Comput. System Sci., Volume 46 (1993) no. 2, pp. 129-154

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