Welcome

pss5 is a general purpose polynomial system solver, under development. While a few solvers are available, this one will incorporate recent mathematical advances on mixed volume computations, sparse condition numbers and condition-metric homotopy.

More precisely, pss5 will find all isolated and non-degenerate roots of a system of n real or complex polynomial equations in n variables. The input looks like this:

[ x1 * x2 - x1 * x3 - x4 + 1,
x2 * x3 - x2 * x4 - x1 + 1,
-x1 * x3 + x3 * x4 - x2 + 1,
x1 * x4 - x2 * x4 - x3 + 1 ]

According to Bézout's theorem, there may exist at most 16 isolated complex roots (x1,x2,x3,x4). This count also includes roots "at infinity". Most of those may be artifacts that exist independent of the coefficients.

A sharper count was given by Bernstein, Kushnirenko and Khovanskii (1976). To each equation, they associate its support, that is the set of exponent vectors that appear in the equation. In the example, the support of the first equation is given by the points [1,1,0,0], [1,0,1,0], [0,0,0,1] and [0,0,0,0]. The Newton polytope associated to this equation is the set of convex linear combinations of those points. The root count is n! times the "mixed volume" associated to this 4-tuple of supports. For this example, the root count is 12.

As polynomial systems grow, there may be an exponential gap between the mixed volume and the Bézout bound. Pre-release 1 of pss5 features a new algorithm for mixed volume computation. It will also produce the "mixed cells", which provide a starting point for a technique known as "nonlinear homotopy".

Pre-release 2 will be able to use homotopy to solve a random system with the same supports. This is an oportunity to incorporate recent mathematical results on path-following to the software. For more details, see the bibliography in my book, and also the notes of the Santaló Summer School on Recent Advances in Real Complexity and Computation.

The actual release will find non-degenerate roots and (HOPEFULLY) use rigorous mathematics to detect if certain homotopy paths lead to degenerate roots, roots at infinity or higher dimensional components.

Current status: pre-release 1 to be available in may.