On the numerical solution of polynomial systems: path-following in toric manifolds

Newton iteration is the archetypal local algorithm for finding isolated, non-degenerate solutions of a system of non-linear equations. By local, it is understood that a good enough approximation of the solution is known. Newton iterates of this approximate solution will then converge quadratically to the exact solution. In this context, it is essential to be able to determine whether an approximation to a solution is close enough. The quadratic convergence theorem in [Sma86] provided an effective criterion, in terms of an invariant that can be evaluated from the available data (the supposed approximate solution) for systems of analytic equations. This and similar results are known as α–theory. Besides giving a local criterion of quadratical convergence for Newton iteration, α–theory can be used in some global settings. For instance, it is possible to bound the complexity of certain homotopy algorithms for solving systems of polynomial equations, or to give lower bounds for root separation. This theory is well-developed for systems of polynomial equations defined on a linear or projective space. I contend that the generalization of α–theory to systems of non-linear equations in more general manifolds. is important and practically relevant. The generalization to certain toric varieties is carried in [Mal01], and numerical results demonstrate its usefulness. Let f be an univariate polynomial, and let ζ be an isolated root of f . One can define different condition numbers for univariate polynomials, according to the ‘space’ the roots are supposed to belong. Classically, old-fashioned numerical analysts consider that roots belong to the complex space C, while a most modern treatment allows for the roots to belong to projective space P. In this paper, we argue that the roots should belong to some toric variety. Example: The table below shows the condition number of a few polynomial systems. The first two examples are borrowed from the PoSSo test suite. The first column shows the normalized condition number. Using a known linear programming technique to scale individual variables, it is possible to dramatically improve the condition number. However, those techniques may be sensitive to small perturbations of the polynomial. The third example is the equation for period–3 orbits of a certain dynamical system. In this case, rescaling variables does not help too much. The last column in the table shows the toric condition number, as explained in this paper. This technique is robust and the toric condition number is stable with respect to perturbations. It is also invariant under rescaling of the variables.