Computing totally real hyperplane sections and linear series on algebraic curves

TL;DR

This paper develops algorithms to determine the existence of totally real hyperplane sections and linear series on real algebraic curves, translating geometric problems into real root counting tasks and providing computational solutions.

Contribution

It introduces new algorithms for deciding the existence of totally real divisors and hyperplane sections on algebraic curves, advancing computational methods in real algebraic geometry.

Findings

Algorithms successfully solve examples of the problem.

Results compare favorably with known bounds for degrees.

Provides computational tools for real algebraic geometry problems.

Abstract

Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may ask whether the corresponding linear series contains an effective divisor with totally real support. This translates into a particular type of parametrized real root counting problem that we wish to solve exactly. On the other hand, it is known that for a given genus and number of real connected components, any linear series of sufficiently large degree contains a totally real effective divisor. Using the algorithms described in this paper, we solve a number of examples, which we can compare to the best known bounds for the required degree.