Infinitely renormalizable quadratic polynomials have been heavily studied. In the context of quadratic-like renormalization, one may try to prove the existence of a priori bounds, a definite thickness for the annuli corresponding to the renormalizations. It has been shown that a priori bounds imply local connectivity of the Julia set and combinatorial rigidity for the corresponding quadratic polynomial (by M. Lyubich in 1997). In a paper from 2006, J. Kahn showed that infinitely renormalizable quadratic polynomials of bounded primitive type admit a priori bounds. In 2002, H. Inou generalized much of the polynomial-like renormalization theory to polynomials of higher degree with several critical points. I will discuss a generalization of Kahn's theorem to the context of polynomials admitting infinitely many primitive renormalizations of bounded type around each of their critical points. These a priori bounds imply local connectivity and rigidity.