Hilbert's zero theorem

In mathematics, Hilbert's zero theorem creates the central connection between ideals and affine algebraic varieties in classical algebraic geometry . He was proven by David Hilbert . There are several equivalent variants of formulating the zero digit set:

• Consider the polynomial ring defined over a field and be the algebraic closure of . Let polynomials in (where they span an ideal ). A zero of these polynomials is an element from . If every common root of the polynomials of the ideal is also a root of , then there is a natural number such that , that is, there are polynomials such that: ${\displaystyle k[X_{1},\ldots ,X_{n}]}$ ${\displaystyle k}$ ${\displaystyle K}$ ${\displaystyle k}$ ${\displaystyle f,f_{1},\ldots ,f_{m}}$ ${\displaystyle k[X_{1},\ldots ,X_{n}]}$ ${\displaystyle f_{i}}$ ${\displaystyle I}$ ${\displaystyle K^{n}}$ ${\displaystyle f_{1},\ldots ,f_{m}}$ ${\displaystyle I}$ ${\displaystyle f}$ ${\displaystyle r}$ ${\displaystyle f^{r}\in I}$ ${\displaystyle g_{1},\ldots ,g_{m}\in k[X_{1},\ldots ,X_{n}]}$

${\displaystyle f^{r}=g_{1}\cdot f_{1}+\cdot \cdot \cdot +g_{m}\cdot f_{m}}$

• Is an algebraically closed body and a real ideal , so there is a so ${\displaystyle K}$ ${\displaystyle {\mathfrak {a}}\subsetneq K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle x\in K^{n}}$

${\displaystyle f(x_{1},\ldots ,x_{n})=0}$for everyone . ${\displaystyle f\in {\mathfrak {a}}}$

${\displaystyle x}$is a common zero of all elements of . In this formulation it is a far-reaching generalization of the fundamental theorem of algebra . ${\displaystyle {\mathfrak {a}}}$

• If an algebraically closed field and an ideal in , then: ${\displaystyle K}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle K[X_{1},\dots ,X_{n}]}$

${\displaystyle {\sqrt {\mathfrak {a}}}=I(V({\mathfrak {a}}))}$

Here means

• ${\displaystyle {\sqrt {\mathfrak {a}}}}$the radical of , ${\displaystyle {\mathfrak {a}}}$
• ${\displaystyle V({\mathfrak {a}})\subseteq K^{n}}$the set of all common zeros of (as above), and ${\displaystyle {\mathfrak {a}}}$
• ${\displaystyle I(X)}$the ideal of all polynomials that vanish on. ${\displaystyle X\subseteq K^{n}}$

The inclusion is trivial, because every zero of is also the zero of . ${\displaystyle {\sqrt {\mathfrak {a}}}\subset I(V({\mathfrak {a}}))}$ ${\displaystyle f(T)^{r}}$ ${\displaystyle f(T)}$

• Let there be a body and a maximum ideal in . Then the degree of body expansion is finite. ${\displaystyle K}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle A=K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle [A/{\mathfrak {m}}:K]}$
• Each prime ideal from the ring (polynomial ring over a body ) is the intersection of the maximum ideals that it contains. This was later taken as the defining property of the Jacobson ring. ${\displaystyle k[X_{1},\cdot \cdot \cdot ,X_{n}]}$ ${\displaystyle k}$
• Let it be an algebraically closed field and a maximal ideal in . Then for a point ${\displaystyle K}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle {\mathfrak {m}}=(X_{1}-a_{1},\ldots ,X_{n}-a_{n})}$ ${\displaystyle (a_{1},\ldots ,a_{n})\in K^{n}}$

• Let it be a field and a field extension that is finitely generated as - algebra . Then it is finite; in particular, the expansion is algebraic. ${\displaystyle K}$ ${\displaystyle L/K}$ ${\displaystyle K}$ ${\displaystyle [L:K]}$

From Hilbert's zero theorem it follows that the mappings and for an algebraically closed body define a bijective relationship between affine algebraic sets in and radical ideals in . This can be restricted to bijective relationships between irreducible algebraic sets and prime ideals and between points in and maximal ideals. ${\displaystyle V}$ ${\displaystyle I}$ ${\displaystyle K^{n}}$ ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle K^{n}}$

Affine varieties are defined by the ideals and the zeros of define related algebraic sets. The zero theorem then says that every non-empty affine variety has an algebraic point. ${\displaystyle I}$ ${\displaystyle I}$

An effective version was proven by W. Dale Brownawell in 1987 for bodies of characteristic zero and by János Kollár in 1988 for any characteristic. Brownawell gave an upper bound for the degrees of the polynomials (compare the first version above), which depends exponentially on the number of variables . ${\displaystyle g_{i}}$ ${\displaystyle n}$

Web links

Wikiversity: A proof of Hilbert's zero theorem  - course materials

• Terence Tao Hilbert's zero set

Individual evidence

1. ↑ Hilbert, Ueber the full invariant systems, Mathematische Annalen, Volume 42, 1893, pp. 313–337
2. ↑ Formulation of the theorem in V. Danilov, Hilbert's zeros theorem , Encyclopedia of Mathematics, Springer
3. ^ Jacobson Ring, Encyclopedia of Mathematics, Springer