Multi-point Padè for the study of phase transitions: from the Ising model to lattice QCD

The Bielefeld Parma collaboration has recently put forward a method to investigate the QCD phase diagram based on the computation of Taylor series coefficients at both zero and imaginary values of the baryonic chemical potential. The method is based on the computation of multi-point Pad\'e approximants. We review the methodological aspects of the computation and, in order to gain confidence in the approach, we report on the application of the method to the two-dimensional Ising model (probably the most popular arena for testing tools in the study of phase transitions). Besides showing the effectiveness of the multi-point Pad\'e approach, we discuss what these results can suggest in view of further progress in the study of the QCD phase diagram. We finally report on very preliminary results in which we look for Pad\'e approximants at different temperatures and fixed values of the (imaginary) baryonic chemical potential.