Optimal transport between algebraic hypersurfaces

TL;DR

This paper introduces a novel optimal transport framework for deforming algebraic hypersurfaces, defining a new inner Wasserstein distance that is complete, geodesic, and induces a smooth Kähler metric with applications in polynomial zeroes and condition numbers.

Contribution

It develops a measure-theoretic optimal transport approach for algebraic hypersurfaces, introducing a finer Wasserstein distance with a smooth Kähler structure, combining complex geometry and optimal transport techniques.

Findings

Inner Wasserstein distance is complete and geodesic.

Distance is induced by a smooth Kähler metric of Weil-Petersson type.

Applications include regularity of polynomial zeroes and condition number analysis.

Abstract

What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces. First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. This distance is finer than the Fubini-Study one. The innner Wasserstein distance is complete and geodesic: geodesics corresponds to optimal deformations of one algebraic hypersurface into another one. Outside the discriminant this distance is induced by a smooth Riemannian metric, which is the real part of an explicit Hermitian structure. Moreover, this Hermitian structure is K\"ahler and the corresponding metric is of Weil-Petersson type. To prove these results we develop new techniques, which combine complex and symplectic geometry with optimal transport, and which we expect to be relevant on their own. We discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.