Hilbert's theorem on zeros

Hilbert's theorem on zeroes ( Hubert's theorem on roots , in many languages, including sometimes in Russian, often use the original German name Nullstellensatz , which translates as the “theorem on zeros”) - a theorem establishing the fundamental relationship between geometry and algebra . The use of this relationship is the basis of algebraic geometry .

This theorem connects the concept of an algebraic set with the concept of an ideal in a ring of polynomials over an algebraically closed field . First proved by David Hilbert ( Math. Ann. 1893, Bd 42, S. 313-373) and named in his honor.

Content

Formulation

Let be $k$ ${\ displaystyle k}$ - an arbitrary field (for example, a field of rational numbers ), $K$ ${\ displaystyle K}$ - an algebraically closed extension of this field (for example, a field of complex numbers ). Will consider $K[x_{1},\ldots ,x_{n}]$ ${\ displaystyle K [x_ {1}, \ ldots, x_ {n}]}$ - polynomial ring from $n$ ${\ displaystyle n}$ variables with coefficients in the field $K$ ${\ displaystyle K}$ , let be $I$ ${\ displaystyle I}$ - ideal in this ring. Algebraic set ${\hbox{V}}(I)$ ${\ displaystyle {\ hbox {V}} (I)}$ defined by this ideal consists of all points $x=(x_{1},\dots ,x_{n})\in K^{n}$ ${\ displaystyle x = (x_ {1}, \ dots, x_ {n}) \ in K ^ {n}}$ such that $f(x)=0$ ${\ displaystyle f (x) = 0}$ for anyone $f\in I$ ${\ displaystyle f \ in I}$ . Hilbert's theorem on zeros states that if some polynomial $p\in k[x_{1},\dots ,x_{n}]$ ${\ displaystyle p \ in k [x_ {1}, \ dots, x_ {n}]}$ vanishes on the set ${\hbox{V}}(I)$ ${\ displaystyle {\ hbox {V}} (I)}$ that is, if $p(x)=0$ ${\ displaystyle p (x) = 0}$ for all $x\in V(I)$ ${\ displaystyle x \ in V (I)}$ then there is a natural number $r$ ${\ displaystyle r}$ such that $p^{r}\in I$ ${\ displaystyle p ^ {r} \ in I}$ .

The immediate consequence is the following "weak form of Hilbert's theorem on zeros": if $I$ ${\ displaystyle I}$ is a proper ideal in the ring $K[x_{1},\dots ,x_{n}]$ ${\ displaystyle K [x_ {1}, \ dots, x_ {n}]}$ then ${\hbox{V}}(I)$ ${\ displaystyle {\ hbox {V}} (I)}$ cannot be an empty set , that is, there exists a common zero for all the polynomials of a given ideal (indeed, otherwise the polynomial $p(x)=1$ ${\ displaystyle p (x) = 1}$ has roots everywhere on ${\hbox{V}}(I)$ ${\ displaystyle {\ hbox {V}} (I)}$ therefore, his degree belongs $I$ ${\ displaystyle I}$ ). This circumstance gave the name of the theorem. The general case can be derived from the “weak form” with the help of the so-called Rabinovich trick . Assumption that the field $K$ ${\ displaystyle K}$ is algebraically closed, essentially: elements of a proper ideal $(x^{2}+1)$ ${\ displaystyle (x ^ {2} +1)}$ at $\mathbb {R} [x]$ ${\ displaystyle \ mathbb {R} [x]}$ do not have a common zero.

Using the standard terminology of commutative algebra , Hilbert's theorem on zeros can be formulated as follows: for each ideal $J$ ${\ displaystyle J}$ fair formula

{\hbox{I}}({\hbox{V}}(J))={\sqrt {J}}

{\ displaystyle {\ hbox {I}} ({\ hbox {V}} (J)) = {\ sqrt {J}}}

Where ${\sqrt {J}}$ ${\ displaystyle {\ sqrt {J}}}$ - radical ideal $J$ ${\ displaystyle J}$ , but ${\hbox{I}}(U)$ ${\ displaystyle {\ hbox {I}} (U)}$ Is an ideal consisting of all polynomials that are zero on the set $U$ ${\ displaystyle U}$ .

It follows from this that operations ${\hbox{I}}$ ${\ displaystyle {\ hbox {I}}}$ and ${\hbox{V}}$ ${\ displaystyle {\ hbox {V}}}$ set a bijective , reversing order on inclusion correspondence between algebraic sets in $K^{n}$ ${\ displaystyle K ^ {n}}$ and radical ideals in $K[x_{1},\ldots ,x_{n}]$ ${\ displaystyle K [x_ {1}, \ ldots, x_ {n}]}$ .

Projective version of Nullstellensatz

There is also a correspondence between homogeneous ideals in a ring of polynomials and algebraic sets in a projective space , called projective Nullstellensatz . Let be $R=K[x_{1},\dots ,x_{n}]$ ${\ displaystyle R = K [x_ {1}, \ dots, x_ {n}]}$ , $R_{d}$ ${\ displaystyle R_ {d}}$ - many homogeneous polynomials of degree $d$ ${\ displaystyle d}$ . Then

R_{+}=\bigoplus _{d\geq 1}R_{d}

{\ displaystyle R _ {+} = \ bigoplus _ {d \ geq 1} R_ {d}}

is called a maximal homogeneous ideal . As in the affine case, we introduce the notation: for the subset $S\subseteq \mathbb {P} ^{n}$ ${\ displaystyle S \ subseteq \ mathbb {P} ^ {n}}$ and homogeneous ideal $I$ ${\ displaystyle I}$ let be

{\begin{aligned}\operatorname {I} _{\mathbb {P} ^{n}}(S)&=\{f\in R_{+}|f(x)=0\;\forall x\in S\},\\\operatorname {V} _{\mathbb {P} ^{n}}(I)&=\{x\in \mathbb {P} ^{n}|f(x)=0\;\forall f\in I\}.\end{aligned}}

{\ displaystyle {\ begin {aligned} \ operatorname {I} _ {\ mathbb {P} ^ {n}} (S) & = \ {f \ in R _ {+} | f (x) = 0 \; \ forall x \ in S \}, \\\ operatorname {V} _ {\ mathbb {P} ^ {n}} (I) & = \ {x \ in \ mathbb {P} ^ {n} | f (x ) = 0 \; \ forall f \ in I \}. \ End {aligned}}}

Recall that $f$ ${\ displaystyle f}$ is not a function on the projective space, but from the homogeneity of this polynomial it follows that the set of points with homogeneous coordinates $x$ ${\ displaystyle x}$ , in which $f(x)=0$ ${\ displaystyle f (x) = 0}$ defined correctly. Now, for an arbitrary homogeneous ideal $I\in R_{+}$ ${\ displaystyle I \ in R _ {+}}$ right

{\sqrt {I}}=\operatorname {I} _{\mathbb {P} ^{n}}(\operatorname {V} _{\mathbb {P} ^{n}}(I)).

{\ displaystyle {\ sqrt {I}} = \ operatorname {I} _ {\ mathbb {P} ^ {n}} (\ operatorname {V} _ {\ mathbb {P} ^ {n}} (I)) .}

Literature

Atia M. , MacDonald I. Introduction to commutative algebra. - M: World, 1972
Van der Warden B. L. Algebra. - M .: Science, 1976.
Prasolov V.V. Polynomials. - M .: MTSNMO , 1999. ISBN 5-900916-32-4 .
Hartshorn R. Algebraic geometry. - M .: Mir, 1970

Hilbert's theorem on zeros

Content

Formulation

Projective version of Nullstellensatz

Literature

See also

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