Diophantine Equations, Hilbert Series, and Undecidable Spaces

Abstract

Summary In 1979 Jan-Erik Roos published a fascinating paper [22] making explicit connections between the Poincare-Betti series of loop spaces or local rings and the Hilbert series of finitely presented graded Hopf algebras. This paper develops a way to model equations within the category of graded Hopf algebras. Combining this with Roos' work, we obtain loop spaces and local rings whose series reflect the solution sets of arbitrary equations. By Matiyasevic's negative solution to Hilbert's tenth problem ([15], [16]), there is no algorithm for deciding in general whether or not a given equation has solutions. For us, one consequence is that no algorithm exists to decide whether a given finitely presented graded algebra is generic (see [3]). As to there is no finite procedure for evaluating whether an arbitrary ring's Poincare series equals a given series, even though by [10] the sequence of coefficients is recursive. Likewise, given two finite simply-connected CW complexes, there is no guaranteed method to tell, in general, whether their loop spaces have the same rational homotopy type. At the same time, Matiyasevic showed that there exist equations, i.e., polynomial equations for which neither the existence nor the non-existence of integer solutions may be proved in a given axiomatization of arithmetic. We obtain undecidable local rings, undecidable spaces, and undecidable topological maps. These concepts will be made precise in Section 4. A few other consequences of the link between equations and Hilbert series may be listed. First, a graded algebra (resp. loop space or local ring) exists whose Hilbert (resp. Poincare) series radius of convergence may be proved to be a transcendental number. Also, the Hilbert series can represent a transcendental function which solves no algebraic differential equation. Lastly, we prove a stability theorem for Diophantine solution sets of bounded complexity.