The Chevalley–Warning theorem
This file contains a proof of the Chevalley–Warning theorem.
Throughout most of this file, K
denotes a finite field
and q
is notation for the cardinality of K
.
Main results
- Let
f
be a multivariate polynomial in finitely many variables (X s
,s : σ
) such that the total degree off
is less than(q-1)
times the cardinality ofσ
. Then the evaluation off
on all points ofσ → K
(akaK^σ
) sums to0
. (sum_mv_polynomial_eq_zero
) - The Chevalley–Warning theorem (
char_dvd_card_solutions
). Letf i
be a finite family of multivariate polynomials in finitely many variables (X s
,s : σ
) such that the sum of the total degrees of thef i
is less than the cardinality ofσ
. Then the number of common solutions of thef i
is divisible by the characteristic ofK
.
Notation
K
is a finite fieldq
is notation for the cardinality ofK
σ
is the indexing type for the variables of a multivariate polynomial ring overK
The Chevalley–Warning theorem.
Let (f i)
be a finite family of multivariate polynomials
in finitely many variables (X s
, s : σ
) over a finite field of characteristic p
.
Assume that the sum of the total degrees of the f i
is less than the cardinality of σ
.
Then the number of common solutions of the f i
is divisible by p
.
The Chevalley–Warning theorem.
Let f
be a multivariate polynomial in finitely many variables (X s
, s : σ
)
over a finite field of characteristic p
.
Assume that the total degree of f
is less than the cardinality of σ
.
Then the number of solutions of f
is divisible by p
.
See char_dvd_card_solutions_family
for a version that takes a family of polynomials f i
.