Section: New Results
Solving Systems over the Reals and Applications
Answering connectivity queries in real algebraic sets
A roadmap for a semi-algebraic set
Polynomial optimization and semi-definite programming
In [6], we describe our freely distributed Maple library spectra, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities, a fundamental object in effective real algebraic geometry and polynomial optimization, with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.
The positive semidefinite rank of a convex body
The Complexity of an Adaptive Subdivision Method for Approximating Real Curves
In [14] we present the first complexity analysis of the algorithm by Plantinga and Vegter for approximating real implicit curves and surfaces. This approximation algorithm certifies the topological correctness of the output using both subdivision and interval arithmetic. In practice, it has been seen to be quite efficient; our goal is to quantify this efficiency. We focus on the subdivision step (and not the approximation step) of the Plantinga and Vegter algorithm. We begin by extending the subdivision step to arbitrary dimensions. We provide a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and the bit complexity for the construction. Then, we use continuous amortization to derive adaptive bounds on the complexity of the subdivided region. We also provide examples showing our bounds are tight.