In a 1996 paper, Andrew Sommese and Charles Wampler began developing a new area, ``Numerical Algebraic Geometry'', which would bear the same relation to ``Algebraic Geometry'' that ``Numerical Linear Algebra'' bears to ``Linear Algebra''.
To approximate all isolated solutions of polynomial systems, numerical path following techniques have been proven reliable and efficient during the past two decades. In the nineties, homotopy methods were developed to exploit special structures of the polynomial system, in particular its sparsity. For sparse systems, the roots are counted by the mixed volume of the Newton polytopes and computed by means of polyhedral homotopies.
In Numerical Algebraic Geometry we apply and integrate homotopy continuation methods to describe solution components of polynomial systems. In particular, our algorithms extend beyond just finding isolated solutions to also find all positive dimensional solution sets of polynomial systems and to decompose these into irreducible components. These methods can be considered as symbolic-numeric, or perhaps rather as numeric-symbolic, since numerical methods are applied to find integer results, such as the dimension and degree of solution components, and via interpolation, to produce symbolic results in the form of equations describing the irreducible components.
Applications from mechanical engineering motivated the development of Numerical Algebraic Geometry. The performance of our software on several test problems illustrates the effectiveness of the new methods.
In A. Dickenstein and I.Z. Emiris (Eds.), Solving Polynomial Equations: Foundations, Algorithms, and Applications. Volume 14 of Algorithms and Computation in Mathematics, Springer-Verlag, pages 339-392, 2005.