 Regular perturbation of a simple algebraic root

If $P({\lambda }_0,0)=0$ and $P_{\lambda }({\lambda }_0,0)\ne 0$, the implicit function theorem gives a local root branch. For analytic coefficients it has an ordinary power series in the perturbation parameter. Its corrections follow by substituting that series into the polynomial and comparing coefficients. A vanishing leading root may require its first nonzero correction to capture a slow exponential mode.