Chevalley–Warning theorem

Chevalley–Warning theorem 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 Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x).

Contents 1 Statement of the theorems 2 Proof of Warning's theorem 3 Artin's conjecture 4 The Ax–Katz theorem 5 See also 6 References 7 External links Statement of the theorems Let {displaystyle mathbb {F} } be a finite field and {displaystyle {f_{j}}_{j=1}^{r}subseteq mathbb {F} [X_{1},ldots ,X_{n}]} be a set of polynomials such that the number of variables satisfies {displaystyle n>sum _{j=1}^{r}d_{j}} where {displaystyle d_{j}} is the total degree of {displaystyle f_{j}} . The theorems are statements about the solutions of the following system of polynomial equations {displaystyle f_{j}(x_{1},dots ,x_{n})=0quad {text{for}},j=1,ldots ,r.} Chevalley–Warning theorem states that the number of common solutions {displaystyle (a_{1},dots ,a_{n})in mathbb {F} ^{n}} is divisible by the characteristic {displaystyle p} of {displaystyle mathbb {F} } . Or in other words, the cardinality of the vanishing set of {displaystyle {f_{j}}_{j=1}^{r}} is {displaystyle 0} modulo {displaystyle p} . Chevalley's theorem states that if the system has the trivial solution {displaystyle (0,dots ,0)in mathbb {F} ^{n}} , i.e. if the polynomials have no constant terms, then the system also has a non-trivial solution {displaystyle (a_{1},dots ,a_{n})in mathbb {F} ^{n}backslash {(0,dots ,0)}} .

Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since {displaystyle p} is at least 2.

Both theorems are best possible in the sense that, given any {displaystyle n} , the list {displaystyle f_{j}=x_{j},j=1,dots ,n} has total degree {displaystyle 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 + ... + xnan 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 {displaystyle q^{n-d}} solutions where {displaystyle q} is the size of the finite field and {displaystyle d:=d_{1}+dots +d_{r}} . Chevalley's theorem also follows directly from this.

Proof of Warning's theorem Remark: If {displaystyle i

Si quieres conocer otros artículos parecidos a Chevalley–Warning theorem puedes visitar la categoría Diophantine geometry.