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Mutually orthogoval projective and affine spaces

Mark Saaltink

Abstract

A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are generalized in two different ways to spaces of higher dimension. The simpler of the two generalizations admits many solutions, both affine and projective. For the stronger definition, where a line of one space must be an arc in the other, we show the existence of pairs of projective spaces of dimension one less than a prime.

Mutually orthogoval projective and affine spaces

Abstract

A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are generalized in two different ways to spaces of higher dimension. The simpler of the two generalizations admits many solutions, both affine and projective. For the stronger definition, where a line of one space must be an arc in the other, we show the existence of pairs of projective spaces of dimension one less than a prime.
Paper Structure (8 sections, 16 theorems, 39 equations)

This paper contains 8 sections, 16 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Constructions of orthogoval affine spaces
  3. Projective spaces
  4. Bounds
  5. Other generalizations
  6. Hyperplane orthogoval and askew spaces
  7. Half-dimension orthogoval
  8. Conclusions

Key Result

Lemma 3

Let $\mathbb{F} = \mathbb{F}_q$ be a finite field and $p \in \mathbb{F}[x]$ be any polynomial. Then there are $a, b \in \mathbb{F}$ such that $p(x) + ax + b$ has no roots in $\mathbb{F}$.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Definition 4
  • Theorem 5
  • proof
  • Corollary 6
  • Lemma 7
  • proof
  • ...and 23 more