Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Extremal and Algorithmic Algebraic Geometry by J. Maurice ROJAS, Texas A&M University Abstract: Over certain fields, such as the reals or the p-adic rationals, one can show that the topology of algebraic sets depends more on the combinatorics of the exponent vectors than on the degrees of the underlying equations. Descartes' Rule -- a 17th century result involving one real polynomial in one real variable -- was the first result in this direction. We discuss analogous theorems for arbitrary systems of equations, over the reals and p-adics, including new counter-examples to Kushnirenko's Conjecture from Fewnomial Theory. These quantitative results motivate the question of whether we can find significant algorithmic speed-ups for real (as opposed to complex) polynomial system solving. We then detail analogous speed-ups for deciding the existence of solutions to certain sparse polynomial systems, over the reals and p-adics. Quantum complexity, and results related to Carmichael numbers, make surprise appearances in the p-adic case.