Editing Chinese remainder theorem (section)

===Hermite interpolation===
[[Hermite interpolation]] is an application of the Chinese remainder theorem for univariate polynomials, which may involve moduli of arbitrary degrees (Lagrange interpolation involves only moduli of degree one).

The problem consists of finding a polynomial of the least possible degree, such that the polynomial and its first [[derivative]]s take given values at some fixed points.

More precisely, let <math>x_1, \ldots, x_k</math> be <math>k</math> elements of the ground [[field (mathematics)|field]] <math>K,</math> and, for <math>i=1,\ldots, k,</math> let <math>a_{i,0}, a_{i,1}, \ldots, a_{i,r_i-1}</math> be the values of the first <math>r_i</math> derivatives of the sought polynomial at <math>x_i</math> (including the 0th derivative, which is the value of the polynomial itself). The problem is to find a polynomial <math>P(X)</math> such that its ''j''{{hairsp}}th derivative takes the value <math>a_{i,j} </math> at <math>x_i,</math> for <math>i=1,\ldots,k</math> and <math>j=0,\ldots,r_j.</math>

Consider the polynomial
:<math>P_i(X) = \sum_{j=0}^{r_i - 1}\frac{a_{i,j}}{j!}(X - x_i)^j.</math>
This is the [[Taylor polynomial]] of order <math>r_i-1</math> at <math>x_i</math>, of the unknown polynomial <math>P(X).</math> Therefore, we must have
:<math>P(X)\equiv P_i(X) \pmod {(X-x_i)^{r_i}}.</math>

[[Converse (logic)|Conversely]], any polynomial <math>P(X) </math> that satisfies these <math>k</math> congruences, in particular verifies, for any <math>i=1, \ldots, k</math>
:<math>P(X)= P_i(X) +o(X-x_i)^{r_i-1} </math> 
therefore <math>P_i(X)</math> is its Taylor polynomial of order <math> r_i - 1</math> at <math>x_i</math>, that is, <math>P(X)</math> solves the initial Hermite interpolation problem.
The Chinese remainder theorem asserts that there exists exactly one polynomial of degree less than the sum of the <math>r_i,</math> which satisfies these <math>k</math> congruences.

There are several ways for computing the solution <math>P(X).</math> One may use the method described at the beginning of {{slink||Over univariate polynomial rings and Euclidean domains}}. One may also use the constructions given in {{slink||Existence (constructive proof)}} or {{slink||Existence (direct proof)}}.