Nonrealizability proofs in computational geometry

Abstract

This paper deals with a class of computational problems in real algebraic geometry. We introduce the concept of final polynomials as a systematic approach to prove nonrealizability for oriented matroids and combinatorial geometries. Hilbert's Nullstellensatz and its real analogue imply that an abstract geometric object is either realizable or it admits a final polynomial. This duality has first been applied by Bokowski in the study of convex polytopes [7] and [11], but in these papers the resulting final polynomials were given without their derivations. It is the objective of the present paper to fill that gap and to describe an algorithm for constructing final polynomials for a large class of nonrealizable chirotopes. We resolve a problem posed in [10] by proving that not every realizable simplicial chirotope admits a solvability sequence. This result shows that there is no easy combinatorial method for proving nonrealizability and thus justifies our final polynomial approach. © 1990 Springer-Verlag New York Inc.