  We develop a collection of numerical algorithms which connect ideas from
polyhedral geometry and algebraic geometry. The first algorithm we develop
functions as a numerical oracle for the Newton polytope of a hypersurface and
is based on ideas of Hauenstein and Sottile. Additionally, we construct a
numerical tropical membership algorithm which uses the former algorithm as a
subroutine. Based on recent results of Esterov, we give an algorithm which
recursively solves a sparse polynomial system when the support of that system
is either lacunary or triangular. Prior to explaining these results, we give
necessary background on polytopes, algebraic geometry, monodromy groups of
branched covers, and numerical algebraic geometry.

Algebraic Geometry