Regeneration, local dimension, and applications in numerical algebraic geometry

Algorithms in the field of numerical algebraic geometry provide numerical methods for computing and manipulating solution sets of polynomial systems. One of the main algorithms in this field is the computation of the numerical irreducible decomposition. This algorithm has three main parts: computing a witness superset, filtering out the junk points to create a witness set, and decomposing the witness set into irreducible components. New and efficient algorithms are presented in this thesis to address the first two parts, namely regeneration and a local dimension test. Regeneration is an equation-by-equation solving method that can be used to efficiently compute a witness superset for a polynomial system. The local dimension test algorithm presented in this thesis is a numerical-symbolic method that can be used to compute the local dimension at an approximated solution to a polynomial system. This test is used to create an efficient algorithm that filters out the junk points. The algorithms presented in this thesis are applied to problems arising in kinematics and partial differential equations.