Summary: The naive perturbation expansion for many-fermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has, to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion self-energy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, J. Feldman, E. Trubowitz and the author have shown sufficient differentiability to solve the FSR equation for a class of models with a non-nested, non-spherical Fermi surface.
In this paper, the author first motivates the problem and gives a definition of FSR, and then describes the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the four-point function. They imply that only ladder diagrams can give singular contributions to the four-point function.