# 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.
- 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.

- 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.

- 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.

- 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.