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Parseval-Rayleigh identities for homogeneous complete intersections

Karim Alexander Adiprasito, Ryoshun Oba, Stavros Argyrios Papadakis, Vasiliki Petrotou

TL;DR

The paper develops Parseval-Rayleigh identities for the residue map of homogeneous complete intersections over fields of positive characteristic, connecting socle-level contractions to a $p$-th power sum over monomials. It proves these identities via a key ideal membership and a Frobenius-compatible contraction lemma, and then derives differential identities that translate the combinatorial Parseval framework into statements about generic complete intersections. A central outcome is the p-anisotropy result and the consequent Lefschetz-type properties, culminating in a conceptual proof that generic homogeneous complete intersections have the Strong Lefschetz Property in characteristic $2$. These results deepen the connection between algebraic identities in positive characteristic and Lefschetz-type phenomena, with implications for Hodge-Riemann-type pairings and related combinatorial conjectures.

Abstract

We prove, in any positive characteristic, Parseval-Rayleigh identities for the residue map of a homogeneous complete intersection. As an application, we give a conceptual proof of the folklore fact that generic homogeneous complete intersections have the Strong Lefschetz Property over any field of characteristic 2.

Parseval-Rayleigh identities for homogeneous complete intersections

TL;DR

The paper develops Parseval-Rayleigh identities for the residue map of homogeneous complete intersections over fields of positive characteristic, connecting socle-level contractions to a -th power sum over monomials. It proves these identities via a key ideal membership and a Frobenius-compatible contraction lemma, and then derives differential identities that translate the combinatorial Parseval framework into statements about generic complete intersections. A central outcome is the p-anisotropy result and the consequent Lefschetz-type properties, culminating in a conceptual proof that generic homogeneous complete intersections have the Strong Lefschetz Property in characteristic . These results deepen the connection between algebraic identities in positive characteristic and Lefschetz-type phenomena, with implications for Hodge-Riemann-type pairings and related combinatorial conjectures.

Abstract

We prove, in any positive characteristic, Parseval-Rayleigh identities for the residue map of a homogeneous complete intersection. As an application, we give a conceptual proof of the folklore fact that generic homogeneous complete intersections have the Strong Lefschetz Property over any field of characteristic 2.

Paper Structure

This paper contains 12 sections, 8 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Statement
  3. Acknowledgements.
  4. Notation
  5. Parseval-Rayleigh Identities
  6. A key ideal membership
  7. A useful lemma
  8. Proof of Theorem \ref{['thm:Parseval']}
  9. Differential identities, $p$-anisotropy and Lefschetz property for generic complete intersections
  10. Generic complete intersection reduction
  11. Differential identities
  12. $p$-anisotropy, Lefschetz property

Key Result

Theorem 1.1

For any $w \in R_s$ we have where ${\mathcal{M}}_s$ denotes the set of all degree $s$ monic monomials of $R$.

Theorems & Definitions (20)

  • Theorem 1.1: Parseval-Rayleigh identities
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:Parseval']}
  • Definition 4.1
  • ...and 10 more