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 - an arbitrary field (for example, a field of rational numbers ), - an algebraically closed extension of this field (for example, a field of complex numbers ). Will consider - polynomial ring from variables with coefficients in the field , let be - ideal in this ring. Algebraic set defined by this ideal consists of all points such that for anyone . Hilbert's theorem on zeros states that if some polynomial vanishes on the set that is, if for all then there is a natural number such that .
The immediate consequence is the following "weak form of Hilbert's theorem on zeros": if is a proper ideal in the ring then cannot be an empty set , that is, there exists a common zero for all the polynomials of a given ideal (indeed, otherwise the polynomial has roots everywhere on therefore, his degree belongs ). 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 is algebraically closed, essentially: elements of a proper ideal at 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 fair formula
Where - radical ideal , but Is an ideal consisting of all polynomials that are zero on the set .
It follows from this that operations and set a bijective , reversing order on inclusion correspondence between algebraic sets in and radical ideals in .
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 , - many homogeneous polynomials of degree . Then
is called a maximal homogeneous ideal . As in the affine case, we introduce the notation: for the subset and homogeneous ideal let be
Recall that 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 , in which defined correctly. Now, for an arbitrary homogeneous ideal right
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