The totally real divisor bound for real algebraic curves

Motivated by the study of nonnegative polynomials on projective real algebraic curves and by symmetric tensor decompositions with nonnegative coefficients, we study the totally real divisor bound.

This is the smallest integer N such that every conjugation-invariant divisor class of degree at least N is represented by a sum of points in the real locus of the curve.

We show that N does not depend only on the topological properties of the curve, but also on the metric properties of the Abel-Jacobi embedding of the curve inside its Jacobian.

In particular, we show that N can be lower bounded by the ratio between the volume of real Jacobian and the length of the real curve in the Bergman metric. This is a joint work with M. Kummer and D. Plaumann.