We present a class of nonlinear (polynomial) models that are discriminatively trained to directly map from the word content in a query-document or document-document pair to a ranking score. Dealing with polynomial models on word features is computationally challenging. We propose a low rank (but diagonal preserving) representation of our polynomial models to induce feasible memory and computation requirements. We provide an empirical study on retrieval tasks based on Wikipedia documents, where we obtain state-of-the-art performance while providing realistically scalable methods.