Computing All Space Curve Solutions of Polynomial Systems by Polyhedral Methods

Abstract

A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.

Pruning Algorithms for Pretropisms of Newton Polytopes

“…The n-body problem [23] is a classical problem from celestial dynamics that states that the acceleration due to Newtonian gravity can be found by solving a system of equations (9). These equations can be turned into a polynomial system by clearing the denominators.…”

“…For polynomial systems with sufficiently generic coefficients, every tropism is also a pretropism. See [9] for an example. Related Work.…”