2013 SIAM Conference on Applied Algebraic Geometry Part of MS49 Algorithms in Real Algebraic Geometry and its Applications - Part II of III Khovanskii-Rolle Continuation for Real Solutions Abstract. Over 30 years ago, Askold Khovanskii used a multivariate generalization of Rolle's Theorem to give a bound on the number of positive solutions to a system of polynomial equations that depends only on the number of terms appearing in the equations, and not on their degree. Reflecting this dependence, this class of bounds are called fewnomial bounds. Khovanskii's proof was revisited by Bihan and Sottile, who gave a refined bound that is asymptotically sharp, in a certain sense. Subsequently, we observed that Khovanskii's use of Rolle's Theorem can form the basis of a continuation algorithm for numerically computing the real solutions to a system of fewnomials. This Khovanskii-Rolle algorithm finds the real solutions by following real arcs, and avoids computing complex solutions to the fewnomial system. Consequently its complexity depends upon the fewnomial bound and not on the number of complex solutions. This talk will explain this Khovanskii-Rolle algorithm and describe some technical challenges that are encountered in its implementation. This represents joint work with Dan Bates. Authors Frank Sottile, Texas A&M University, USA, sottile@math.tamu.edu Daniel Bates, Colorado State University, USA, bates@math.colostate.edu