Multivariate polynomial ring over a field - Revision history

Vipul: 4 revisions

2008-05-12T16:27:31Z

4 revisions

← Older revision	Revision as of 16:27, 12 May 2008
(No difference)

Vipul: /* Automorphisms and endomorphisms */

2008-02-08T23:38:45Z

Automorphisms and endomorphisms

← Older revision		Revision as of 23:38, 8 February 2008
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	===As a filtered ring===		===As a filtered ring===

	The multivariate polynomial ring <math>k[x_1,x_2,\ldots,x_n]</math>		The multivariate polynomial ring <math>k[x_1,x_2,\ldots,x_n]</math> has a filtration, where the <math>d^{th}</math> level of the filtration comprises all polynomials of degree at most <math>d</math>. The endomorphisms for this filtration are thus endomorphisms that are degree-preserving.

			* In particular, the endomorphisms preserve the first level of the filtration, i.e. they send linear polynomials to linear polynomials. This shows that the monoid of endomorphisms is naturally identified with the affine maps from <math>k^n</math> to <math>k^n</math> i.e. maps that preserve collinearity and ratios of distances along a line.
			* The automorphisms are identified with the general affine group <math>GA(n,k)</math>, the group of collinearity-cum-ratio preserving bijective maps from <math>k^n</math> to <math>k^n</math>.

			===As a graded ring===

			The multivariate polynomial ring <math>k[x_1,x_2,\ldots,x_n]</math> has a gradation, where the <math>d^{th}</math> graded component is generated over <math>k</math> by the monomials of total degree <math>d</math>. The endomorphisms that preserve this gradation must send homogeneous linear polynomials to homogeneous linear polynomials.

			* In particular, the endomorphism monoid is the multiplicative monoid of the matrix ring <math>M(n,k)</math>
			* The automorphism group is the general linear group <math>GL(n,k)</math>

Vipul at 23:29, 8 February 2008

2008-02-08T23:29:03Z

← Older revision		Revision as of 23:29, 8 February 2008
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	{{further\|[[Max-spectrum of multivariate polynomial ring over a field]], [[max-spectrum of multivariate polynomial ring over an algebraically closed field]]}}		{{further\|[[Max-spectrum of multivariate polynomial ring over a field]], [[max-spectrum of multivariate polynomial ring over an algebraically closed field]]}}

			==Automorphisms and endomorphisms==

			{{further\|[[endomorphism monoid of multivariate polynomial ring over a field]], [[automorphism group of multivariate polynomial ring over a field]]}}

			===As an algebra over the base field===

			The multivariate polynomial ring <math>k[x_1,x_2,\ldots,x_n]</math> can be viewed as a <math>k</math>-algebra. The endomorphisms with respect to this structure are the endomorphisms that fix the elements of <math>k</math>.

			* The endomorphisms are in bijective correspondence with sequences of polynomials <math>p_1,p_2,\ldots,p_n</math>. Here <math>p_i</math> is the polynomial to which <math>x_i</math> goes under the endomorphisms. Composition of endomorphisms is by composing the polynomials.
			* The automorphisms are the multiplicative subgroup of this monoid of endomorphisms described above. In general, there is no easy or direct description by which one can test whether a given endomorphism is an automorphism.

			===As a filtered ring===

			The multivariate polynomial ring <math>k[x_1,x_2,\ldots,x_n]</math>

Vipul: /* Spectrum */

2008-02-08T23:21:07Z

Spectrum

← Older revision		Revision as of 23:21, 8 February 2008
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	==Spectrum==		==Spectrum==

			{{further\|[[Spectrum of multivariate polynomial ring over a field]], [[spectrum of multivariate polynomial ring over an algebraically closed field]]}}

	===Max-spectrum===		===Max-spectrum===

	{{further\|[[Max-spectrum of multivariate polynomial ring over a field]], [[max-spectrum of multivariate polynomial ring over an algebraically closed field]]}}		{{further\|[[Max-spectrum of multivariate polynomial ring over a field]], [[max-spectrum of multivariate polynomial ring over an algebraically closed field]]}}

Vipul: New page: ==Definition== The '''multivariate polynomial ring over a field''' is defined as a polynomial ring in (finitely) many variables over a field. If we denote the underlying field by $k<...$

2008-02-08T23:20:23Z

New page: ==Definition== The '''multivariate polynomial ring over a field''' is defined as a polynomial ring in (finitely) many variables over a field. If we denote the underlying field by <math>k<...

New page

==Definition==

The '''multivariate polynomial ring over a field''' is defined as a polynomial ring in (finitely) many variables over a field. If we denote the underlying field by <math>k</math>, and the variables by <math>x_1,x_2,\ldots,x_n</math>, then the polynomial ring is denoted <math>k[x_1,x_2,\ldots,x_n]</math>.

==Properties==

The multivariate polynomial ring over a field is a [[unique factorization domain]] as well as a [[Noetherian domain]]. When there is more than one variable, it is not a [[Euclidean domain]] or a [[principal ideal domain]].

==Extra structure==

{{further|[[multivariate polynomial ring#extra structure]]}}

The multivariate polynomial ring <math>k[x_1,x_2,\ldots,x_n]</math> can be viewed as a <math>k</math>-algebra. It also has the following additional structures:

* [[Graded ring]] structure: In particular, it is a graded <math>k</math>-algebra where the <math>d^{th}</math> component is the vector space generated by monomials of degree <math>d</math>
* [[Filtered ring]] structure: In particular, it is a filtered <math>k</math>-algebra where the <math>d^{th}</math> filtration is the subspace of polynomials of degree at most <math>d</math>

==Ideals and quotients==

Quotients of the polynomial ring in <math>n</math> variables, are precisely the same as algebras generated by <math>n</math> elements as <math>k</math>-algebras. The structure of these is in general fairly complicated.

==Spectrum==

===Max-spectrum===

{{further|[[Max-spectrum of multivariate polynomial ring over a field]], [[max-spectrum of multivariate polynomial ring over an algebraically closed field]]}}

Multivariate polynomial ring over a field - Revision history

Vipul: 4 revisions

Vipul: /* Automorphisms and endomorphisms */

Vipul at 23:29, 8 February 2008

Vipul: /* Spectrum */

Vipul: New page: ==Definition== The '''multivariate polynomial ring over a field''' is defined as a polynomial ring in (finitely) many variables over a field. If we denote the underlying field by k<...

Vipul: New page: ==Definition== The '''multivariate polynomial ring over a field''' is defined as a polynomial ring in (finitely) many variables over a field. If we denote the underlying field by $k<...$