Computing Puiseux Series for Algebraic Surfaces
This work provides a computational tool for algebraic geometers studying singularities of surfaces, but the method is preliminary and limited to specific cases.
The paper presents an algorithmic method for computing Puiseux series expansions for algebraic surfaces, using polyhedral methods to find tropisms. The approach yields exact representations for solution sets of the cyclic n-roots problem for n = m^2, confirming a result by Backelin.
In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface. Our approach starts with a polyhedral method to compute cones of normal vectors to the Newton polytopes of the given polynomial system that defines the surface. If as many vectors in the cone as the dimension of the surface define an initial form system that has isolated solutions, then those vectors are potential tropisms for the initial term of the Puiseux series expansion. Our preliminary methods produce exact representations for solution sets of the cyclic $n$-roots problem, for $n = m^2$, corresponding to a result of Backelin.