Chevalley–Warning theorem

Let $\backslash mathbb\{F\}$ be a finite field and $\backslash \{f\_j\backslash \}\_\{j=1\}^r\backslash subseteq\backslash mathbb\{F\}[X\_1,\backslash ldots,X\_n]$ be a set of polynomials such that the number of variables satisfies

:$n>\backslash sum\_\{j=1\}^r\; d\_j$

where $d\_j$ is the total degree of $f\_j$. The theorems are statements about the solutions of the following system of polynomial equations

:$f\_j(x\_1,\backslash dots,x\_n)=0\backslash quad\backslash text\{for\}\backslash ,\; j=1,\backslash ldots,\; r.$

• The Chevalley–Warning theorem states that the number of common solutions $(a\_1,\backslash dots,a\_n)\; \backslash in\; \backslash mathbb\{F\}^n$ is divisible by the characteristic $p$ of $\backslash mathbb\{F\}$. Or in other words, the cardinality of the vanishing set of $\backslash \{f\_j\backslash \}\_\{j=1\}^r$ is $0$ modulo $p$.
• The Chevalley theorem states that if the system has the trivial solution $(0,\backslash dots,0)\; \backslash in\; \backslash mathbb\{F\}^n$, that is, if the polynomials have no constant terms, then the system also has a non-trivial solution $(a\_1,\backslash dots,a\_n)\; \backslash in\; \backslash mathbb\{F\}^n\; \backslash backslash\; \backslash \{(0,\backslash dots,0)\backslash \}$.

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 $f\_j\; =\; x\_j,\; j=1,\backslash dots,n$ has total degree $n$ and only the trivial solution. Alternatively, using just one polynomial, we can take f₁ to be the degree n polynomial given by the norm of x₁a₁ + ... + x_na_n where the elements a form a basis of the finite field of order pⁿ.

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 $q^\{n-d\}$ solutions where $q$ is the size of the finite field and $d\; :=\; d\_1\; +\; \backslash dots\; +\; d\_r$. Chevalley's theorem also follows directly from this.

Remark:{{Cite web |url=https://math.stackexchange.com/questions/1352150/number-of-solutions-to-polynomials-in-finite-fields |title=Number of Solutions to Polynomials in Finite Fields |website=StackExchange}} If

:$\backslash sum\_\{x\backslash in\backslash mathbb\{F\}\}x^i=0$

so the sum over $\backslash mathbb\{F\}^n$ of any polynomial in $x\_1,\backslash ldots,x\_n$ of degree less than $n(q-1)$ also vanishes.

The total number of common solutions modulo $p$ of $f\_1,\; \backslash ldots,\; f\_r\; =\; 0$ is equal to

:$\backslash sum\_\{x\backslash in\backslash mathbb\{F\}^n\}(1-f\_1^\{q-1\}(x))\backslash cdot\backslash ldots\backslash cdot(1-f\_r^\{q-1\}(x))$

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.

It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin 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, named after James Ax and Nicholas Katz, determines more accurately a power $q^b$ of the cardinality $q$ of $\backslash mathbb\{F\}$ 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

: $\backslash frac\{n\; -\; \backslash sum\_j\; d\_j\}\{d\}.$

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. Namely, the same power of $q$ divides each of these algebraic integers.

• Combinatorial Nullstellensatz

{{reflist}}

• {{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | editor1-last=Lang | editor1-first=Serge. | editor2-last=Tate | editor2-first=John | editor2-link=John Tate (mathematician) | title=Collected papers | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90686-7 | mr=671416 | year=1982}}
• {{citation| last=Ax | first=James | authorlink=James Ax | year=1964 | title=Zeros of polynomials over finite fields | journal=American Journal of Mathematics | volume=86 | pages=255–261 | mr=0160775 | doi=10.2307/2373163}}
• {{citation| last=Chevalley | first=Claude | authorlink=Claude Chevalley | year=1935 | title=Démonstration d'une hypothèse de M. Artin | language=French| journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=11 | pages=73–75 | zbl=0011.14504 |jfm=61.1043.01| doi=10.1007/BF02940714}}
• {{citation| last=Katz | first=Nicholas M. | authorlink=Nicholas Katz | year=1971 | title=On a theorem of Ax | journal=Amer. J. Math. | volume=93 | issue=2 | pages=485–499 | doi=10.2307/2373389}}
• {{citation| last=Warning | first=Ewald | year=1935 | title=Bemerkung zur vorstehenden Arbeit von Herrn Chevalley|language=German | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=11 | pages=76–83 | zbl=0011.14601 |jfm=61.1043.02| doi=10.1007/BF02940715}}
• {{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | year=1973 | title=A course in arithmetic | pages=[https://archive.org/details/courseinarithmet00serr/page/5 5–6] | isbn=0-387-90040-3 | url-access=registration | url=https://archive.org/details/courseinarithmet00serr/page/5 }}

• {{cite web |url=https://mathoverflow.net/q/178318 |title=Proofs of the Chevalley-Warning theorem}}

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Category:Finite fields

Category:Diophantine geometry

Category:Theorems in algebra