Chevalley-Warning Theorem

Let 𝔽q be the finite fieldMathworldPlanetmath of q elements with characteristicPlanetmathPlanetmath p. Let fi(x1,,xn), i=1,2,,r, be polynomialPlanetmathPlanetmath of n variables over 𝔽q. If n>i=1rdeg(fi), then the number of solutions over 𝔽q to the system of equations

f1(x1,x2,,xn) =0
f2(x1,x2,,xn) =0
fr(x1,x2,,xn) =0

is divisible by p. In particular, if none of the polynomials f1, f2,,fr have constant term, then there are at least p solutions.

Title Chevalley-Warning Theorem
Canonical name ChevalleyWarningTheorem
Date of creation 2013-03-22 17:46:52
Last modified on 2013-03-22 17:46:52
Owner kshum (5987)
Last modified by kshum (5987)
Numerical id 6
Author kshum (5987)
Entry type Theorem
Classification msc 12E20