Editing Polynomial ring (section)

==Several indeterminates over a field==
Polynomial rings in several variables over a field are fundamental in [[invariant theory]] and [[algebraic geometry]]. Some of their properties, such as those described above can be reduced to the case of a single indeterminate, but this is not always the case. In particular, because of the geometric applications, many interesting properties must be  invariant under [[affine transformation|affine]] or [[projective transformation|projective]] transformations of the indeterminates. This often implies that one cannot select one of the indeterminates for a recurrence on the indeterminates.

[[Bézout's theorem]], [[Hilbert's Nullstellensatz]] and [[Jacobian conjecture]] are among the most famous properties that are specific to multivariate polynomials over a field.

=== Hilbert's Nullstellensatz ===
{{Main|Hilbert's Nullstellensatz}}
The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by [[David Hilbert]], which extends to the multivariate case some aspects of the [[fundamental theorem of algebra]]. It is foundational for [[algebraic geometry]], as establishing a strong link between the algebraic properties of <math>K[X_1, \ldots, X_n]</math> and the geometric properties of [[algebraic varieties]], that are (roughly speaking) set of points defined by [[implicit equation|implicit polynomial equations]].

The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz.

The first version generalizes the fact that a nonzero univariate polynomial has a [[complex number|complex]] zero if and only if it is not a constant. The statement is: ''a set of polynomials {{mvar|S}} in <math>K[X_1, \ldots, X_n]</math> has a common zero in an [[algebraically closed field]] containing {{mvar|K}}, if and only if'' {{math|1}} ''does not belong to the [[ideal (ring theory)|ideal]] generated by {{mvar|S}}, that is, if'' {{math|1}} ''is not a [[linear combination]] of elements of {{mvar|S}} with polynomial coefficients''.

The second version generalizes the fact that the [[irreducible polynomial|irreducible univariate polynomial]]s over the complex numbers are [[associate elements|associate]] to a polynomial of the form <math>X-\alpha.</math> The statement is: ''If {{mvar|K}} is algebraically closed, then the [[maximal ideal]]s of <math>K[X_1, \ldots, X_n]</math> have the form <math>\langle X_1 - \alpha_1, \ldots, X_n - \alpha_n \rangle.</math>''

===Bézout's theorem===
{{main|Bézout's theorem}}
Bézout's theorem may be viewed as a multivariate generalization of the version of the [[fundamental theorem of algebra]] that asserts that a univariate polynomial of degree {{mvar|n}} has {{mvar|n}} complex roots, if they are counted with their multiplicities.

In the case of [[bivariate polynomial]]s, it states that two polynomials of degrees {{mvar|d}} and {{mvar|e}} in two variables, which have no common factors of positive degree, have exactly {{mvar|de}} common zeros in an [[algebraically closed field]] containing the coefficients, if the zeros are counted with their multiplicity and include the [[point at infinity|zeros at infinity]].

For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with [[homogeneous polynomial]]s, and consider zeros in a [[projective space]]. In this context, a ''projective zero'' of a homogeneous polynomial <math>P(X_0, \ldots, X_n)</math> is, up to a scaling, a {{math|(''n'' + 1)}}-[[tuple]] <math>(x_0, \ldots, x_n)</math>  of elements of {{mvar|K}} that is different from {{math|(0, …, 0)}}, and such that <math>P(x_0, \ldots, x_n) = 0 </math>. Here, "up to a scaling" means that <math>(x_0, \ldots, x_n)</math> and <math>(\lambda x_0, \ldots, \lambda x_n)</math> are considered as the same zero for any nonzero <math>\lambda\in K.</math> In other words, a zero is a set of [[homogeneous coordinates]] of a point in a projective space of dimension {{mvar|n}}.

Then, Bézout's theorem states: Given {{mvar|n}} homogeneous polynomials of degrees <math>d_1, \ldots, d_n</math> in {{math|''n'' + 1}} indeterminates, which have only a finite number of common projective zeros in an [[algebraically closed extension]] of {{mvar|K}}, the sum of the [[multiplicity (mathematics)#Intersection multipliicty|multiplicities]] of these zeros is the product <math>d_1 \cdots d_n.</math>

===Jacobian conjecture===
{{main|Jacobian conjecture}}
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