Abstract Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Grobner basis for the poly- ¨ nomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Grobner bases leads to more efficient solvers in many ¨ cases. We present a novel basis sampling scheme that we evaluate on a number of problems