Chevalley–Warning theorem
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In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are
quasi-algebraically closed field In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebra ...
s .


Statement of the theorems

Let \mathbb be a finite field and \_^r\subseteq\mathbb _1,\ldots,X_n/math> be a set of polynomials such that the number of variables satisfies :n>\sum_^r d_j where d_j is the total degree of f_j. The theorems are statements about the solutions of the following system of polynomial equations :f_j(x_1,\dots,x_n)=0\quad\text\, j=1,\ldots, r. * The ''Chevalley–Warning theorem'' states that the number of common solutions (a_1,\dots,a_n) \in \mathbb^n is divisible by the characteristic p of \mathbb. Or in other words, the cardinality of the vanishing set of \_^r is 0 modulo p. * The ''Chevalley theorem'' states that if the system has the trivial solution (0,\dots,0) \in \mathbb^n, that is, if the polynomials have no constant terms, then the system also has a non-trivial solution (a_1,\dots,a_n) \in \mathbb^n \backslash \. Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since p is at least 2. Both theorems are best possible in the sense that, given any n, the list f_j = x_j, j=1,\dots,n has total degree n and only the trivial solution. Alternatively, using just one polynomial, we can take ''f''1 to be the degree ''n'' polynomial given by the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of ''x''1''a''1 + ... + ''x''''n''''a''''n'' where the elements ''a'' form a basis of the finite field of order ''p''''n''. Warning proved another theorem, known as Warning's second theorem, which states that if the system of polynomial equations has the trivial solution, then it has at least q^ solutions where q is the size of the finite field and d := d_1 + \dots + d_r. Chevalley's theorem also follows directly from this.


Proof of Warning's theorem

''Remark:'' If i then :\sum_x^i=0 so the sum over \mathbb^n of any polynomial in x_1,\ldots,x_n of degree less than n(q-1) also vanishes. The total number of common solutions modulo p of f_1, \ldots, f_r = 0 is equal to :\sum_(1-f_1^(x))\cdot\ldots\cdot(1-f_r^(x)) because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials f_i is less than ''n'' then this vanishes by the remark above.


Artin's conjecture

It is a consequence of Chevalley's theorem that finite fields are
quasi-algebraically closed In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebrai ...
. This had been conjectured by
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn's theorem.


The Ax–Katz theorem

The Ax–Katz theorem, named after
James Ax James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in ...
and
Nicholas Katz Nicholas Michael Katz (born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at P ...
, determines more accurately a power q^b of the cardinality q of \mathbb dividing the number of solutions; here, if d is the largest of the d_j, then the exponent b can be taken as the ceiling function of : \frac. 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. Namely, the same power of q divides each of these algebraic integers.


See also

*
Combinatorial Nullstellensatz In additive number theory and combinatorics, a restricted sumset has the form :S=\, where A_1,\ldots,A_n are finite nonempty subsets of a field ''F'' and P(x_1,\ldots,x_n) is a polynomial over ''F''. If P is a constant non-zero function, for ...


References

* * * * * *


External links

* {{DEFAULTSORT:Chevalley-Warning theorem Finite fields Diophantine geometry Theorems in algebra