Does there exists some simple criteria to know when the primitive of a rational function of CC[z]

Jupellodseple804

Jupellodseple804

Answered question

2022-02-17

Does there exists some simple criteria to know when the primitive of a rational function of C[z] is still a rational function?
In fact my question is more about the stability of this property. Let P and Q two co' polynomials and let A and B two co' polynomials such that
AB=(PQ)=PQPQQ2
Then considering a pertubation Aξ of A (in the sense the roots of Aξ converge to the one of A with the same multiplicity as ξ goes to zero.): does AξB admit a primitive which is rational function?

Answer & Explanation

Kwame Malone

Kwame Malone

Beginner2022-02-18Added 5 answers

One simple criterion is the following. If the denominator of f has squarefree factorization Q1Q22Qnn then f has a partial fraction expansion P1Q1++PnQnn, which is the derivative of a rational function each PQi is QW(P,Q,(Q2),,(Qi1)), where W denotes the Wronskian. Recall that the squarefree factorization of a polynomial over a field of characteristic 0 may be quickly computed by gcds.

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