Let be a field, let be a polynomial of degree . Given a polynomial of degree less than and an integer , we want to find a rational function with satisfying
We recall that a rational function is said to be in canonical form if is monic and if . Clearly, every rational function has a unique canonical form.
The following Theorem 5:
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Next, we prove . So we consider a solution to such that the fraction is in canonical form. There exists such that we have . Since holds, we can apply Proposition 3. So, let be such that
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If then we have . If then we have .
We prove that holds. If then , and thus , since holds, which implies . If , we can apply Proposition 3 again. Then, there exists such that and both hold. It follows that we have and . Hence, we deduce
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Marc Moreno Maza