Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

Some Speed-Ups and Speed Limits for Real Algebraic Geometry

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel in that it is stated in terms of the volumes of certain polytopes and, for a large class of inputs, beats the best previous bounds by a factor exponential in the number of variables. (2) A new algorithm for approximating the real roots of certain sparse polynomial systems. Two features of our algorithm are (a) arithmetic complexity polylogarithmic in the degree of the underlying complex variety (as opposed to the super-linear dependence in earlier algorithms) and (b) a simple and efficient generalization to certain univariate exponential sums. (3) Detecting whether a real algebraic surface (given as the common zero set of some input straight-line programs) is not smooth can be done in polynomial time within the classical Turing model (resp. BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of Complexity. More typos are corrected, and a new section is added where the bounds here are compared to an earlier result of Benedetti, Loeser, and Risler. The LaTeX source needs the ajour.cls macro file to compil

Counting Solutions of a Polynomial System Locally and Exactly

We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system $f_1=\cdots=f_n=0$, with $f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a polydisc $\mathbf{\Delta}\subset\mathbb{C}^n$, our method aims to certify the existence of $k$ solutions (counted with multiplicity) within the polydisc. In case of success, it yields the correct result under guarantee. Otherwise, no information is given. However, we show that our algorithm always succeeds if $\mathbf{\Delta}$ is sufficiently small and well-isolating for a $k$-fold solution $\mathbf{z}$ of the system. Our analysis of the algorithm further yields a bound on the size of the polydisc for which our algorithm succeeds under guarantee. This bound depends on local parameters such as the size and multiplicity of $\mathbf{z}$ as well as the distances between $\mathbf{z}$ and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots in $\mathbf{\Delta}$ of the original system to the problem of solving a truncated system of degree $k$. In particular, if the multiplicity $k$ of $\mathbf{z}$ is small compared to the total degrees of the polynomials $f_i$, our method considerably improves upon known complete and certified methods. For the special case of a bivariate system, we report on an implementation of our algorithm, and show experimentally that our algorithm leads to a significant improvement, when integrated as inclusion predicate into an elimination method