Computing all Space Curve Solutions of Polynomial Systems by Polyhedral Methods
This work provides a method for computing all space curve solutions of polynomial systems, which is a fundamental problem in algebraic geometry and its applications, though the contribution is incremental as it extends existing polyhedral methods to handle a specific difficult case.
The paper proposes a hybrid symbolic-numeric method to compute Puiseux series expansions for all space curve solutions of polynomial systems, addressing the difficult case where leading powers lie in the relative interior of a higher-dimensional cone of the tropical prevariety. The method resolves this case using polyhedral end games, and the authors show that this difficult case does not occur for generic coefficients.
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.