 Jupellodseple804

2022-02-17

Does there exists some simple criteria to know when the primitive of a rational function of $\mathbb{C}\left[z\right]$ 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
$\frac{A}{B}=\left(\frac{P}{Q}{\right)}^{\prime }=\frac{{P}^{\prime }Q-P{Q}^{\prime }}{{Q}^{2}}$
Then considering a pertubation ${A}^{\xi }$ of A (in the sense the roots of ${A}^{\xi }$ converge to the one of A with the same multiplicity as $\xi$ goes to zero.): does $\frac{{A}^{\xi }}{B}$ admit a primitive which is rational function? Kwame Malone

One simple criterion is the following. If the denominator of f has squarefree factorization ${Q}_{1}{Q}_{2}^{2}\dots {Q}_{n}^{n}$ then f has a partial fraction expansion $\frac{{P}_{1}}{{Q}_{1}}+\dots +\frac{{P}_{n}}{{Q}_{n}^{n}}$, which is the derivative of a rational function $⇔$ each $\frac{P}{{Q}^{i}}$ is $⇔Q\mid W\left(P,{Q}^{\prime },{\left({Q}^{2}\right)}^{\prime },\dots ,{\left({Q}^{i-1}\right)}^{\prime }\right)$, 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.