Comptes Rendus
no. 9
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 Bourgain1
1 Institute for Advanced Study, Princeton, NJ 08540, USA
Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 627-631.

On démontre que la corrélation sur {0,1}n de la fonction parité et un polynôme p(x1,…,xn)∈Z[X1,…,Xn](modq), q un entier impair donné et p(X) de degré d arbitraire mais fixé, est exponentiellement petite en n pour n→∞. 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(x1,…,xn)∈Z[X1,…,Xn](modq), q a fixed odd number and p(X) of degree d arbitrary but fixed, is exponentially small in n as n→∞. An application to circuit complexity, from where the problem originates, is given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.03.008
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/

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[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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