Chevalley-Warning theorem

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In mathematics, Chevalley's theorem on solutions of polynomial equations over a finite field F with q elements, q a power of the prime number p, states that for a polynomial

P(X₁, ..., X_N)

of total degree d, with

d < N,

the number M of solutions of

P(X₁, ..., X_N) = 0

in F^N is divisible by p. As a corollary, when P is a homogeneous polynomial of degree d, there is at least one solution other than

(0, ..., 0).

This was proved in 1936 by Claude Chevalley, in response to a question of Emil Artin. There is an elementary proof using the p-th power map. See also quasi-algebraic closure.

Warning's generalisation is for several polynomials P_j and simultaneous equations, under the condition that the sum of the total degrees d_j is less than N. It equally has the corollary that when the polynomials are homogeneous, there is a non-zero solution.

The Ax-Katz theorem (here Katz is to Warning as Ax is to Chevalley) determines more accurately a power q^b dividing the number M of solutions: here, if d is the largest of the d_j , the exponent b can be taken as the ceiling function of

(N − Σ d_j)/d.

The Ax-Katz result has an interpretation in étale cohomology, as a divisibility result for the (reciprocals of) the zeroes and poles of the local zeta-function. The same power of q divides each of these algebraic integers.

[edit] References

• C. Chevalley, C. "Démonstration d'une hypothèse de M. Artin." Abhand. Math. Sem. Hamburg 11, 73-75, 1936.
• E. Warning, Bemerkung zur vorstehenden Arbeit von Herr Chevalley (Abh. Math. Sem. Univ. Hamburg, Vol. 11, 1936, pp. 76-83)
• J. Ax, Zeros of polynomials over finite fields, Amer. J. Math., 86(1964), 255-261
• N. M. Katz, On a theorem of Ax, Amer. J. Math., 93(1971), 485-499.