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On the Schmidt and analytic ranks for trilinear forms

Karim Adiprasito, David Kazhdan, Tamar Ziegler

TL;DR

The paper studies and interrelates Schmidt rank, slice rank, and analytic rank for multilinear polynomials, focusing on trilinear forms. It proves that, for trilinear forms, Schmidt rank and analytic rank are essentially proportional over finite fields, via both rough (geometric-counting) and sharp (tangent-space) bounds. A key contribution is adapting Schmidt's complex-field bound to finite characteristic when d=3, establishing $r(P) le 2g$ with $g$ the codimension of the zero-set $Z_P$, and deriving quantitative relationships between ranks. The work also connects these rank notions to broader conjectures about inverses of Gowers norms and rank stability across base-fields, providing pathways for applications in additive combinatorics and algebraic geometry. Overall, it offers a unified, field-sensitive framework linking algebraic geometry, invariant theory, and rank notions for multilinear forms.

Abstract

We discuss relations between different notions of ranks for multilinear forms. In particular we show that the Schmidt and the analytic ranks for trilinear forms are essentially proportional.

On the Schmidt and analytic ranks for trilinear forms

TL;DR

The paper studies and interrelates Schmidt rank, slice rank, and analytic rank for multilinear polynomials, focusing on trilinear forms. It proves that, for trilinear forms, Schmidt rank and analytic rank are essentially proportional over finite fields, via both rough (geometric-counting) and sharp (tangent-space) bounds. A key contribution is adapting Schmidt's complex-field bound to finite characteristic when d=3, establishing with the codimension of the zero-set , and deriving quantitative relationships between ranks. The work also connects these rank notions to broader conjectures about inverses of Gowers norms and rank stability across base-fields, providing pathways for applications in additive combinatorics and algebraic geometry. Overall, it offers a unified, field-sensitive framework linking algebraic geometry, invariant theory, and rank notions for multilinear forms.

Abstract

We discuss relations between different notions of ranks for multilinear forms. In particular we show that the Schmidt and the analytic ranks for trilinear forms are essentially proportional.

Paper Structure

This paper contains 10 sections, 16 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['2']}
  3. Rough bound
  4. A lemma on the codimension
  5. A proof of the rough bound
  6. Proof of Theorem \ref{['maint']}(1)
  7. The adaptation of the Schmidt's result for fields of finite characteristic in the case when $d=3$.
  8. Linear Algebra
  9. Transcendence degree
  10. Proof of Theorem \ref{['Schmidt(f)']}

Key Result

Lemma 1.5

In the case when ${\mathbf k} =\mathbb F _q$ we have $A_{{\mathbf k} , \psi }(P)=\frac{| {\mathbf Z} _P (\mathbb F_q)|}{q^{\sum _{i=2}^d \mathrm{dim}(V_i)}}$.

Theorems & Definitions (48)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Claim 1.4
  • Lemma 1.5
  • proof
  • Corollary 1.6
  • Conjecture 1.7: d
  • Remark 1.8
  • Theorem 1.9: Dersken
  • ...and 38 more