We show that non-representable oriented matroids always possess a final polynomial as a consequence of an appropriate real version of Hilbert's Nullstellensatz.Expand

We provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space.Expand

We introduce the concept of final polynomials as a systematic approach to prove nonrealizability for oriented matroids and combinatorial geometries.Expand

The classification of the 1296 (simplicial) 3-spheres with nine vertices into polytopal and nonpolytopal spheres, started earlier, is completed here.Expand