Dixon Beats Gr¨ obner: “Almost Linear” Polynomial Equations Arising in GPS Systems and in Nash Equilibria Robert H. Lewis Fordham University, New York, NY 10458, USA Dedicated to the 100th anniversary of the publication by Arthur Lee Dixon of The Eliminant of Three Quantics in Two Independent Variables We apply the Dixon-EDF resultant method [8] to several sets of multivariate poly- nomial equations. They arise in two applications. The first is GPS, or global positioning systems. We show that a 3D affine transformation problem can be completely solved sym- bolically with Dixon-EDF. Other symbolic techniques failed. One of these systems has 6 equations in 6 variables and 12 parameters. Another has 9 equations in 9 variables and 18 parameters. In both systems, every equation has (total) degree three. See [12]. Secondly, we use Dixon-EDF to solve several sets of equations that arise from the study of Nash equilibria. This is an important topic in economic game theory. We examine the cases of three or four players with two pure strategies each. The latter produces a set of 8 equations with 8 variables and 32 parameters. Then we look at a classic problem due to Nash, simplified three-man poker (with 4 equations, 4 variables, 44 parameters), and lastly at a “cube game” (8, 8, 4). These are found in the book by Sturmfels [15] and the papers by Datta [2],[3]. Apparently we are the first to provide fully symbolic solutions to these games. All of these problems are solvable with Dixon-EDF. Apparently all are intractable with other methods. We report on failed attempts to solve these with Maple12, using both its builtin Gr¨ obner bases command and its implementation of Faugere’s fgb algorithm [5]. There is another common thread in these two apparently disparate subjects: all the equations are of degree one in each variable. That is, in every equation no variable is squared. In only one of the equations is any parameter squared. Indeed, often we find that every equation is of total degree two in the variables. These are in some sense the simplest non-linear equations; we call them “almost linear.” Yet Gr¨ obner bases methods fail repeatedly as Dixon-EDF succeeds. Acknowledgments: Thanks to Bela Palancz for showing me GPS problems, and to Ruchira Datta for showing me Nash equilibria.