Polyhedral Methods in Numerical Algebraic Geometry
For researchers in numerical algebraic geometry, this work offers a preprocessing method that can lower the computational expense of solving polynomial systems with positive-dimensional solution sets.
The paper introduces certificates for algebraic curves based on Puiseux series expansions and tropisms, linking their computation to mixed volumes to reduce the cost of witness set computations in numerical algebraic geometry.
In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.