16 research outputs found
Convex polarities over ordered fields
AbstractWe use tools and methods from real algebraic geometry (spaces of ultrafilters, elimination of quantifiers) to formulate a theory of convexity in KN over an arbitrary ordered field. By defining certain ideal points (which can be viewed as generalizations of recession cones) we obtain a generalized notion of polar set. These satisfy a form of polar duality that applies to general convex sets and does not reduce to classical duality if K is the field of real numbers. As an application we give a partial classification of total orderings of Artinian local rings and two applications to ordinary convex geometry over the real numbers
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape