Vipul: New page: {{integral domain property}} ==Definition== An integral domain $R$ is termed an '''interpolation domain''' if, for any nonnegative integer $n$ , there exist elem...

2009-02-06T02:16:42Z

New page: {{integral domain property}} ==Definition== An integral domain <math>R</math> is termed an '''interpolation domain''' if, for any nonnegative integer <math>n</math>, there exist elem...

New page

{{integral domain property}}

==Definition==

An [[integral domain]] <math>R</math> is termed an '''interpolation domain''' if, for any nonnegative integer <math>n</math>, there exist elements <math>(r_0, r_1, \dots, r_n) \in R^{n+1}</math> giving a bijection between the polynomials in <math>\operatorname{Int}(R)</math> (the [[defining ingredient::ring of integer-valued polynomials]] over <math>R</math>) of degree at most <math>n</math> and the elements of <math>R^{n+1}</math> by:

<math>f \mapsto (f(r_0),f(r_1), \dots, f(r_n))</math>.

In other words, for any tuple <math>(a_0, a_1, \dots, a_n) \in R^{n+1}</math>, there exists a unique <math>f \in \operatorname{Int}(R)</math> such that <math>f(r_i) = a_i</math> for every <math>0 \le i \le n</math>.

==Examples==

* Every [[field]] is an interpolation domain. This can be seen in many ways; one of them is the [[Lagrange interpolation formula]].
* The [[particular example::ring of rational integers]] is an interpolation domain. We can set <math>r_i = i</math> for this ring. In other words, given any sequence of <math>n+1</math> integers, we can find an integer-valued polynomial sending <math>0,1, \dots, n</math> to the elements of that sequence. {{proofat|[[Ring of rational integers is an interpolation domain]]}}

Interpolation domain - Revision history

Vipul: New page: {{integral domain property}} ==Definition== An integral domain R is termed an '''interpolation domain''' if, for any nonnegative integer n, there exist elem...

Vipul: New page: {{integral domain property}} ==Definition== An integral domain $R$ is termed an '''interpolation domain''' if, for any nonnegative integer $n$ , there exist elem...