Constructive Newton–Puiseux Theorem, Sheaf Model of the Separable Closure and Dynamic Evaluation

Abstract: Computing the Puiseux expansions of a plane algebraic curve defined by an affine equation over an algebraically closed field is a an important algorithm in algebraic geometry. This is the so-called Newton–Puiseux Theorem. The termination of this algorithm, however, is usually justified by non-constructive means. By adding a separability condition we obtain a variant of the algorithm, the termination of which is justified constructively in characteristic 0. Furthermore, we present a possible constructive interpretation of the existence of the separable closure of a field by building, in a constructive metatheory, a suitable site model where there is such separable closure. Consequently, one can dispense with the assumption of separable closure and extract computational content from proofs involving this assumption. The theorem of Newton-Puiseux is one example where we use the sheaf model to extract computational content. We then can find Puiseux expansions of an algebraic curve defined over a non-algebraically closed field K of characteristic 0. The expansions are given as a fractional power series over a finite dimensional K-algebra.