An algorithmic reduction to canonical forms for vector bundles on anisotropic conics

TL;DR

This paper presents a polynomial-time algorithm for reducing vector bundle transition matrices on real anisotropic conics to canonical block diagonal forms, generalizing classical results on the Riemann sphere.

Contribution

It introduces a new polynomial complexity algorithm for canonical reduction of vector bundles on anisotropic conics and provides an elementary algebraic proof of their decomposition into indecomposables.

Findings

Algorithm reduces transition matrices to canonical forms efficiently.

Provides an elementary proof of vector bundle decomposition on anisotropic conics.

Method generalizes to anisotropic conics over arbitrary fields.

Abstract

We describe a polynomial complexity algorithm for reducing transition matrices, for vector bundles glued along a clutching-type cover of a real anisotropic conic, to canonical block diagonal forms. This is a generalization, to the real anisotropic form, of the classification of vector bundles on the Riemann sphere by their canonical diagonal forms due to Grothendieck and Birkhoff. To enable our algorithm, we provide an elementary algebraic proof for the result, due to Biswas-Nagaraj and Novakovic, of the decomposition of vector bundles on real anisotropic conics into sums of indecomposable vector bundles of rank at most 2. While our algorithm and our proof of this decomposition focus solely on the setting of a real anisotropic conic, our methods are immediately generalizable to anisotropic conics over arbitrary fields.