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
Bourgain, Jean1
1 Institute for Advanced Study, Princeton, NJ 08540, USA
Comptes Rendus. Mathématique, Tome 340 (2005) no. 9, pp. 627-631

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.

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.

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

Bourgain, Jean 1

1 Institute for Advanced Study, Princeton, NJ 08540, USA 
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Bourgain, Jean. Estimation of certain exponential sums arising in complexity theory. Comptes Rendus. Mathématique, Tome 340 (2005) no. 9, pp. 627-631. doi: 10.1016/j.crma.2005.03.008

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

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

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