This paper presents a program analysis method that generates program invariants involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. Such techniques are able to generate all polynomial invariants for a restricted class of programs. However, there currently lacks approaches to apply prior complete methods to the case of general programs with nested loops, conditional branching, unstructured control flow, etc. This paper bridges that gap. Our approach works uniformly for every loop, regardless of whether the loop is within the class of programs prior methods can solve or not. Instead of restricting the kinds of programs we can handle, our method abstracts every loop into a model that can be solved with prior techniques. Our uniform approach to loops gives us a method that brings to bear prior work on solvable polynomial maps to general programs. While no method can generate all polynomial invariants for arbitrary programs, our method establishes its merit through a monotonicty result. We have implemented our techniques, and tested them on a suite of benchmarks from the literature. Our experiments indicate our techniques show promise on challenging verification tasks requiring powerful non-linear reasoning.