Polynomial system solving: the case of a six-atom molecule

A relatively new branch of computational biology and chemistry has been emerging as an e ort to apply successful paradigms and algorithms from geometry and robot kinematics to predicting the structure of molecules, embedding them in Euclidean space, and nding the energetically favorable con gurations. We illustrate several e cient algebraic algorithms for enumerating all possible conformations of a cyclic molecule and for studying its singular locus. Recent advances in computational algebra are exploited, including distance geometry, sparse polynomial theory based on Newton polytopes, and matrix methods for solving nonlinear multivariate polynomial systems. With respect to the latter, we compare sparse resultants, Bezoutians, and Sylvester resultants in cascade, in terms of performance and numerical stability. Key-words: polynomial equation, sparse resultant, Bezoutian, Sylvester resultant, eigenvalue, singular value, singular locus, geometry of lines, cylclohexane, computational biology and chemistry