Explicit Generation of Integer Solutions via Cy Manifolds

Abstract

Metrics on Calabi-Yau manifolds are used to derive a formula that finds the existence of integer solutions to polynomials. These metrics are derived from an associated algebraic curve, together with its anti-holomorphic counterpart. The integer points in the curve coincide with points on the manifold, and the metric form around these points are used to find their existence. The explicit form of the metrics can be found through a solution to the D-terms in a non-linear sigma model. The metrics on general CY manifolds have recently been computed in [1],[2]. This task was accomplished by using a field theoretic count of classical tree graphs in scalar field theories In prior work, the metrics of these Calabi-Yau manifolds was shown to provide means to finding both solutions to systems of algebraic equations and non-linear partial differential equations [3],[4] (with related work in [5]-[7]). In this work explicit formulae are given that generate the integer solutions to polynomial equations using the metric form of these manifolds. Metric Expansions and Integer Solutions The starting point is the algebraic equation

Cite this paper

@inproceedings{Chalmers2005ExplicitGO, title={Explicit Generation of Integer Solutions via Cy Manifolds}, author={Gordon Chalmers}, year={2005} }