Khovanskii-Rolle continuation for real solutions Frank Sottile Texas A&M University Continuation methods are numerical algorithms which find all solutions to a system of polynomials by numerically tracing curves. Well-known are homptopy methods, where the curves arise from degenerations of the system, connecting solutions to the original system to those for simpler systems. With Dan Bates, we propose a different method which is based on Khovanskii's generalization of Rolle's theorem and the notion of Gale duality for polynomial systems. Unlike homotopy continuation, this algorithm only finds real-number solutions and its complexity depends only on the number of real solutions, and not on the algebraic degree of the equations. In this talk, I will sketch the main ideas in this new algorithm and show how it works in an implementation that we have made which is restricted to Gale-dual systems in the plane.