Theorem

Alexander-Hirschowitz Theorem

Algebra

The Alexander-Hirschowitz theorem describes when a collection of double points in projective space imposes independent conditions on the homogeneous polynomials defining hypersurfaces of a given dimension, generalizing classical polynomial interpolation in several variables to points with higher multiplicities. The theorem gives a complete answer along with a short, known list of exceptional cases where the expected count of independent conditions fails to hold.

Facts

Statement

The Alexander-Hirschowitz theorem shows that a specific collection of k double points in the projective space Pr will impose independent types of conditions on homogenous polynomials and the hypersurface of dimension d with many known lists of exceptions. 1

Classification

Statement Form

Classification Theorem 1

Connections

Has Statement Form

Classification Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Sources

1. Alexander-Hirschowitz theorem (Wikipedia)

Introduction

Quote, Introduction

The Alexander?Hirschowitz theorem shows that a specific collection of k double points in the projective space Pr will impose independent types of conditions on homogenous polynomials

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Alexander-Hirschowitz Theorem

Has Statement Form