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Additive Combinatorics Using Equivariant Cohomology

László M. Fehér, János Nagy

Abstract

We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erdős-Heilbronn conjecture. We generalize a theorem of G. Kós (the Grashopper problem) which in some sense is a simultaneous generalization of the Erdős-Heilbronn conjecture. We also prove a signed version of the Erdős-Heilbronn conjecture and the Grashopper problem. Most identities used are based on calculating the projective degree of an algebraic variety in two different ways.

Additive Combinatorics Using Equivariant Cohomology

Abstract

We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erdős-Heilbronn conjecture. We generalize a theorem of G. Kós (the Grashopper problem) which in some sense is a simultaneous generalization of the Erdős-Heilbronn conjecture. We also prove a signed version of the Erdős-Heilbronn conjecture and the Grashopper problem. Most identities used are based on calculating the projective degree of an algebraic variety in two different ways.

Paper Structure

This paper contains 19 sections, 27 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Known results using equivariant cohomology
  3. New results
  4. Warmup: The Cauchy--Davenport theorem, connections with Lagrange interpolation and the Combinatorial Nullstellensatz
  5. Cohomological proof of the Coefficient Formula
  6. Dias da Silva--Hamidoune theorem revisited
  7. The Sun conjecture
  8. The Grasshopper: a simultaneous generalization of the Erdős--Heilbronn problem
  9. Schubert varieties in the flag manifold and Bruhat restrictions for the grasshopper
  10. The Symplectic case
  11. The small $p$ case
  12. Grasshopper with signs
  13. Concluding Remarks
  14. The Borel--Hirzebruch formula
  15. $\mathop{\mathrm{GL}}\nolimits(n)$
  16. ...and 4 more sections

Key Result

Theorem 1

Let $A \subset \mathbb{F}_{p}$ be a set, such that $|A| = n$ and $1 \leq k \leq n$, and let us use the notation Then $|\bigwedge^kA| \geq \min\{(n-k)k+1, p \}$ holds.

Theorems & Definitions (51)

  • Theorem
  • Lemma
  • Theorem
  • Theorem
  • Definition
  • Theorem
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • ...and 41 more