Ring of integer-valued polynomials - Revision history

Vipul at 17:01, 5 February 2009

2009-02-05T17:01:56Z

← Older revision		Revision as of 17:01, 5 February 2009
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			{{variation of\|polynomial ring}}
	==Definition==		==Definition==

Vipul at 16:13, 1 February 2009

2009-02-01T16:13:41Z

← Older revision		Revision as of 16:13, 1 February 2009
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	* For the [[ring of rational integers]] <math>\mathbb{Z}</math>, the ring of integer-valued polynomials equals the ring generated by binomial polynomials. {{proofat\|[[Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials]]}}		* For the [[ring of rational integers]] <math>\mathbb{Z}</math>, the ring of integer-valued polynomials equals the ring generated by binomial polynomials. {{proofat\|[[Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials]]}}
	* An [[interpolation domain]] is an integral domain for which interpolation using integer-valued polynomials is possible: for any degree <math>n</math>, there exist <math>n + 1</math> points such that the integer-valued polynomials of degree <math>n</math> can be interpolated from any collection of values at those <math>n + 1</math> points.		* An [[interpolation domain]] is an integral domain for which interpolation using integer-valued polynomials is possible: for any degree <math>n</math>, there exist <math>n + 1</math> points such that the integer-valued polynomials of degree <math>n</math> can be interpolated from any collection of values at those <math>n + 1</math> points.

			==As an operator==

			We can view the ''ring of integer-valued polynomials'' as an operator that takes as input an integral domain and outputs another integral domain (Note: Unlike the polynomial ring, this operator is not functorial). We can then ask what properties of the original integral domain continue to hold in the new ring. It turns out that most good properties, such as Noetherianness and unique factorization, do not hold any more, even when the starting ring is as nice as <math>\mathbb{Z}</math>. There are, however, some redeeming features:

			* [[Ring of integer-valued polynomials over normal domain is normal]]

Vipul: New page: ==Definition== Let $R$ be an integral domain and let $K$ be its field of fractions. The '''ring of integer-valued polynomials''' for $R$ , denoted ...

2009-01-24T02:49:59Z

New page: ==Definition== Let <math>R</math> be an integral domain and let <math>K</math> be its field of fractions. The '''ring of integer-valued polynomials''' for <math>R</math>, denoted ...

New page

==Definition==

Let <math>R</math> be an [[integral domain]] and let <math>K</math> be its [[field of fractions]]. The '''ring of integer-valued polynomials''' for <math>R</math>, denoted <math>\operatorname{Int}(R)</math>, is defined as the subset of the [[polynomial ring over a field|polynomial ring]] <math>K[x]</math> comprising those polynomials <math>f</math> such that <math>f(x) \in R</math> whenever <math>x \in R</math>.

==Facts==

* In general, <math>R[x]</math> is a subring of <math>\operatorname{Int}(R)</math>, which in turn is a subring of <math>K[x]</math>.
* When <math>K</math> has characteristic zero, the ring of integer-valued polynomials is contained in the [[ring generated by binomial polynomials]] over <math>R</math>. {{proofat|[[Ring of integer-valued polynomials is contained in ring generated by binomial polynomials]]}}
* For the [[ring of rational integers]] <math>\mathbb{Z}</math>, the ring of integer-valued polynomials equals the ring generated by binomial polynomials. {{proofat|[[Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials]]}}
* An [[interpolation domain]] is an integral domain for which interpolation using integer-valued polynomials is possible: for any degree <math>n</math>, there exist <math>n + 1</math> points such that the integer-valued polynomials of degree <math>n</math> can be interpolated from any collection of values at those <math>n + 1</math> points.

Ring of integer-valued polynomials - Revision history

Vipul at 17:01, 5 February 2009

Vipul at 16:13, 1 February 2009

Vipul: New page: ==Definition== Let R be an integral domain and let K be its field of fractions. The '''ring of integer-valued polynomials''' for R, denoted ...

Vipul: New page: ==Definition== Let $R$ be an integral domain and let $K$ be its field of fractions. The '''ring of integer-valued polynomials''' for $R$ , denoted ...