# Application of Polynomial Homotopy Continuation to Scientific and Engineering Problems

Polynomial systems arise in various fields of science and engineering. In the methodology of homotopy continuation methods we distinguish four components.
1. A homotopy defines the deformation from a general to a special problem, continuation methods track the paths from the solutions of the special problem to the general configuration. Counting the roots mirrors the resolution process of the special problem. Homotopies turn formal root counting methods into effective numerical solvers. The regularity of the solution paths provides a justification for the root count. Thus, we are looking for constructive proofs in algebraic geometry.

2. Symbolic computation is a necessary step in the preparation of the numerical calculations. The crunching of equations consists in extracting the characteristic data to count the roots, followed by the set up of the homotopies that determine the deformations. The goal is to find the numerically most favorable embedding of the problem to enable efficient and reliable approximation of the solutions.

3. Most polynomial systems that arise in practical applications are sparse, in the sense that not all monomials appear with nonzero coefficients. Newton polytopes provide an excellent model to capture and exploit the sparsity. Polyhedral methods construct efficient representations of the polytopes and their subdivisions.

4. To test the practical efficiency of the newly developed methods and to explore new possibilities for improvement, the creation and maintenance of publicly available software is essential. Applications from engineering include mechanical design and control theory.