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Numerical algebraic geometry provides tools for approximating solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial systems that differ only in coefficients, not monomials. This technique is frequently used for solving a parameterized family of polynomial systems at multiple parameter values. This article describes Paramotopy, a parallel, optimized implementation of this technique, making use of the Bertini software package. The novel features of this implementation include allowing for the simultaneous solutions of arbitrary polynomial systems in a parameterized family on an automatically generated or manually provided mesh in the parameter space of coefficients, front ends and back ends that are easily specialized to particular classes of problems, and adaptive techniques for solving polynomial systems near singular points in the parameter space. 

The problem of geolocation of a radiofrequency transmitter via time difference of arrival (TDOA) and frequency difference of arrival (FDOA) is given as a system of polynomial equations. This allows for the use of homotopy continuation-based methods from numerical algebraic geometry. A novel geolocation algorithm employs numerical algebraic geometry techniques in conjunction with the random sample consensus (RANSAC) method. This is all developed and demonstrated in the setting of only FDOA measurements, without loss of generality. Additionally, the problem formulation as polynomial systems immediately provides lower bounds on the number of receivers or measurements required for the solution set to consist of only isolated points. 

Given a polynomial system f, this article provides a general construction for homotopies that yield at least one point of each con- nected component on the set of solutions of f = 0. This algorithmic approach is then used to compute a superset of the isolated points in the image of an algebraic set which arises in many applications, such as com- puting critical sets used in the decomposition of real algebraic sets. An example is presented which demonstrates the efficiency of this approach.