1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

Polynomial systems occur in a wide variety of application domains. Homotopy continuation methods are reliable and powerful methods to compute numerically approximations to all isolated complex solutions. During the last decade considerable progress has been accomplished on exploiting structure in a polynomial system, in particular its sparsity. In this article the structure and design of the software package PHC is described. The main program operates in several modes, is menu driven, and is file oriented. This package features great variety of root-counting methods among its tools. The outline of one black-box solver is sketched, and a report is given on its performance on a large database of test problems. The software has been developed on four different machine architectures. Its portability is ensured by the gnu-ada compiler.

http://eaton.math.rpi.edu/CSUMS/Papers/Homotopy/verschelde.pdf

Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation

The volume and the number of lattice points of the permutohedron Pn are given by certain multivariate polynomials that have remarkable com- binatorial properties. We give 3 dieren t formulas for these polynomials. We also study a more general class of polytopes that includes the permutohedron, the associahedron, the cyclohedron, the Stanley-Pitman polytope, and various generalized associahedra related to wonderful compactications of De Concini- Procesi. These polytopes are constructed as Minkowsky sums of simplices. We calculate their volumes and describe their combinatorial structure. The coef- cien ts of monomials in Vol Pn are certain positive integer numbers, which we call the mixed Eulerian numbers. These numbers are equal to the mixed volumes of hypersimplices. Various specializations of these numbers give the usual Eulerian numbers, the Catalan numbers, the numbers (n+1) n 1 of trees (or parking functions), the binomial coecien ts, etc. We calculate the mixed Eulerian numbers using certain binary trees. Many results are extended to an arbitrary Weyl group.

/pdf/permutohedra-associahedra-and-beyond-44871dvfvj.pdf

Permutohedra, Associahedra, and Beyond

Geometric and solid modelling deal with the representation and manipulation of physical objects. Currently most geometric objects are formulated in terms of polynomial equations, thereby reducing many application problems to manipulating polynomial systems. Solving systems of polynomial equations is a fundamental problem in these geometric computations. The author presents an algorithm for solving polynomial equations. The combination of multipolynomial resultants and matrix computations underlies this efficient, robust and accurate algorithm. >

D. N. Bernshtein

Papers

The number of roots of a system of equations