论文标题
代数超曲面之间的最佳运输
Optimal transport between algebraic hypersurfaces
论文作者
论文摘要
将投射性高表面变形为另一个的最佳方法是什么？在本文中，我们将回答这个问题，该问题采用了度量理论的观点，并引入了复杂的代数投影性超曲面之间的最佳运输问题。 首先，构建了给定程度的高空空间空间中的自然拓扑嵌入到投射空间上的度量空间中。然后，通过约束的动态公式来定义高空曲面之间的最佳传输问题，从而最大程度地减少了完全连续曲线的能量，而曲线位于该嵌入的图像上。以这种方式，引入了均匀多项式的射影空间上的内部瓦斯汀距离。这个距离比Fubini-Study的距离更好。 Innner Wasserstein的距离是完整的，地球：测量学对应于一个代数超表面的最佳变形。在判别之外，这种距离是由光滑的riemannian度量引起的，这是显式遗产结构的真实部分。此外，这种隐士结构是Kähler，相应的指标是Weil-Petersson类型。 为了证明这些结果，我们开发了新技术，这些技术将复杂和符合性的几何形状与最佳运输结合在一起，并且我们希望它们独自一人相关。 我们讨论了有关多元多项式家族的零零和多项式系统求解的条件数量的应用。
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 Kähler 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.
