4400ChevalleyWarning

4400ChevalleyWarning - THE CHEVALLEY-WARNING THEOREM...

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Unformatted text preview: THE CHEVALLEY-WARNING THEOREM (FEATURING...THE ERD ¨ OS-GINZBURG-ZIV THEOREM) PETE L. CLARK 1. The Chevalley-Warning Theorem In this handout we shall discuss a result that was conjectured by Emil Artin in 1935 and proved shortly thereafter by Claude Chevalley. A refinement was given by Artin’s graduate student Ewald Warning, who, as the story goes, was the one whom Artin had intended to prove the theorem before Chevalley came visiting Got- tingen and got Artin to say a little too much about the mathematics his student was working on. One of the charms of the Chevalley-Warning theorem is that it can be stated and appreciated without much motivational preamble. So let’s just jump right in. 1.1. Statement of the theorem(s). Let q = p a be a prime power, and let F q be a finite field of order q . We saw earlier in the course that there exists a finite field of each prime power cardinality. 1 For the reader who is unfamiliar with finite fields, it may be a good idea to just replace F q with F p = Z /p Z on a first reading, and then afterwards look back and see that the assumption of an arbitrary finite field changes nothing. Theorem 1. (Chevalley’s Theorem) Let n , d 1 ,...,r be positive integers such that d 1 + ... + d r < n . For each 1 ≤ i ≤ r , let P i ( t 1 ,...,t n ) ∈ F q [ t 1 ,...,t n ] be a polynomial of total degree d i with zero constant term: P i (0 ,..., 0) = 0 . Then there exists ̸ = x = ( x 1 ,...,x n ) ∈ F n q such that P 1 ( x ) = ... = P r ( x ) = 0 . Exercise 1: Suppose we are given any system of polynomials P 1 ( t ) ,...,P r ( t ) in n variables t 1 ,...,t n with ∑ i deg( P i ) < n . Deduce from Chevalley’s that if there exists at least one x ∈ F n q such that P 1 ( x ) = ... = P r ( x ), then there exists y ̸ = x such that P 1 ( y ) = ... = P r ( y ). (Hint: Make a change of variables to reduce to Chevalley’s theorem.) In other words, Exercise 1 asserts that a system of polynomials in n variables over F q cannot have exactly one common solution, provided the sum of the degrees is less than n . Warning’s theorem gives a generalization: 1 It can be shown that any two finite fields of the same order are isomorphic; indeed this is (literally) a textbook application of the uniqueness of splitting fields of polynomials and can be found in any graduate level algebra text treating field theory. But we don’t need this uniquness statement here. 1 2 PETE L. CLARK Theorem 2. (Warning’s Theorem) Let n , d 1 ,...,r be positive integers such that d 1 + ... + d r < n . For each 1 ≤ i ≤ r , let P i ( t 1 ,...,t n ) ∈ F q [ t 1 ,...,t n ] be a polynomial of total degree d i . Let Z = # { ( x 1 ,...,x n ) ∈ F n q | P 1 ( x 1 ,...,x n ) = ... = P r ( x 1 ,...,x n ) = 0 . } Then Z ≡ 0 (mod p ) ....
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This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at UGA.

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4400ChevalleyWarning - THE CHEVALLEY-WARNING THEOREM...

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