On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials

Reed-Solomon list decoding from a system-theoretic perspective

In this paper, the Sudan-Guruswami approach to list decoding of Reed-Solomon (RS) codes is cast in a system-theoretic framework. With the data, a set of trajectories or time series is associated which is then modeled as a so-called behavior. In this way, a connection is made with the behavioral approach to system theory. It is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the Sudan-Guruswami approach. The concept of "weighted row reduced" is introduced and used to achieve minimality. Two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple's Isolate for polynomial system solving, and CGAL's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011. arXiv admin note: substantial text overlap with arXiv:1010.1386 and arXiv:1103.469