This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010 in Banff, Ontario, Canada. This volume contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and touch upon core topics such as homotopy methods for approximating complex solutions, robust floating point methods for clusters of roots, and speed-ups for counting real solutions. Vital related topics such as circuit complexity, random polynomials over local fields, tropical geometry, and the theory of fewnomials, amoebae, and coamoebae are treated as well. Recent advances on Smale's 17th Problem, which deals with numerical algorithms that approximate a single complex solution in average-case polynomial time, are also surveyed.
Multivariate ultrametric root counting by M. Avendano and A. Ibrahim A parallel endgame by D. J. Bates, J. D. Hauenstein, and A. J. Sommese Efficient polynomial system solving by numerical methods by C. Beltran and L. M. Pardo Symmetric determinantal representation of formulas and weakly skew circuits by B. Grenet, E. L. Kaltofen, P. Koiran, and N. Portier Mixed volume computation in solving polynomial systems by T.-L. Lee and T.-Y. Li A search for an optimal start system for numerical homotopy continuation by A. Leykin Complex tropical localization, and coamoebas of complex algebraic hypersurfaces by M. Nisse Randomization, sums of squares, near-circuits, and faster real root counting by O. Bastani, C. J. Hillar, D. Popov, and J. M. Rojas Dense fewnomials by K. Rusek, J. Shakalli, and F. Sottile The numerical greatest common divisor of univariate polynomials by Z. Zeng