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Separating invariants for two-dimensional orthogonal groups over finite fields

Artem Lopatin, Pedro Antonio Muniz Martins

TL;DR

The article analyzes separating invariants for the modular orthogonal group $O_2^+(F_q)$ acting on $m$-tuples of vectors in a 2D space $V=F_q^2$, deriving a minimal separating set for $F_q[V^m]^{O_2^+(F_q)}$ and an explicit orbit classification. It proves a canonical orbit form (Theorem 2.1) and shows that a compact, explicit set of invariants suffices to separate all distinguishable points (Theorem 3.1), with $sigma_sep=2$ and $beta_sep$ taking values $2$ for $q=2$ and $q-1$ otherwise. The work demonstrates that separating sets can be substantially smaller than generating sets in the modular orthogonal setting and provides quantitative bounds on the minimal separating degree and size via orbit counts. These results offer practical, compact certificates for distinguishing orbits in applications and advance the understanding of modular invariant theory for low-dimensional orthogonal groups over finite fields.

Abstract

We described a minimal separating set for the algebra of $O(F_q)$-invariant polynomial functions of $m$-tuples of two-dimensional vectors over a finite field $F_q$.

Separating invariants for two-dimensional orthogonal groups over finite fields

TL;DR

The article analyzes separating invariants for the modular orthogonal group acting on -tuples of vectors in a 2D space , deriving a minimal separating set for and an explicit orbit classification. It proves a canonical orbit form (Theorem 2.1) and shows that a compact, explicit set of invariants suffices to separate all distinguishable points (Theorem 3.1), with and taking values for and otherwise. The work demonstrates that separating sets can be substantially smaller than generating sets in the modular orthogonal setting and provides quantitative bounds on the minimal separating degree and size via orbit counts. These results offer practical, compact certificates for distinguishing orbits in applications and advance the understanding of modular invariant theory for low-dimensional orthogonal groups over finite fields.

Abstract

We described a minimal separating set for the algebra of -invariant polynomial functions of -tuples of two-dimensional vectors over a finite field .
Paper Structure (9 sections, 4 theorems, 32 equations)

This paper contains 9 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Separating invariants
  3. Vector invariants
  4. Orthogonal invariants
  5. Results
  6. Notations
  7. Classification of ${\rm O}_2^{+}({\mathbb{F}}_q)$-orbits
  8. ${\rm O}_2^{+}({\mathbb{F}}_q)$-invariants
  9. Corollaries

Key Result

lemma 1

Assume that $u,v,w,u',v',w'\in V$ and $\alpha\in{\mathbb{F}}_q^{\times}$.

Theorems & Definitions (19)

  • remark 1
  • remark 2
  • proof
  • lemma 1
  • proof
  • proof
  • proof
  • remark 3
  • lemma 2
  • proof
  • ...and 9 more