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The Vandermonde determinant identity in higher dimension

Itaï Ben Yaacov

Abstract

We generalise the Vandermonde determinant identity to one which tests whether a family of hypersurfaces in $\mathbf{P}^n$ has an unexpected intersection point.

The Vandermonde determinant identity in higher dimension

Abstract

We generalise the Vandermonde determinant identity to one which tests whether a family of hypersurfaces in has an unexpected intersection point.

Paper Structure

This paper contains 3 sections, 12 theorems, 45 equations.

Table of Contents

  1. The linear identity
  2. An asymptotic higher-degree identity
  3. A remark regarding the Macaulay resultant

Key Result

Proposition 1.0

With $\Lambda$ and $M_\Lambda$ as above, we have where $\Lambda'$ is taken with the induced order from $\Lambda$.

Theorems & Definitions (27)

  • Proposition 1.0: Linear dual Vandermonde identity in arbitrary dimension
  • proof
  • Corollary 1.0: Linear Vandermonde identity in arbitrary dimension
  • proof
  • Remark 1.0: Ulysse Serres's « usine à points »
  • proof
  • Definition 2.0
  • Definition 2.0
  • Lemma 2.0
  • proof
  • ...and 17 more