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Nuclei of Normal Rational Curves

Johannes Gmainer, Hans Havlicek

Abstract

A $k$-nucleus of a normal rational curve in PG$(n,F)$ is the intersection over all $k$-dimensional osculating subspaces of the curve ($k\in\{-1,0,...,n-1\}$). It is well known that for characteristic zero all nuclei are empty. In case of characteristic $p>0$ and $# F\geq n$ the number of non-zero digits in the representation of $n+1$ in base $p$ equals the number of distinct nuclei. An explicit formula for the dimensions of $k$-nuclei is given for $# F\geq k+1$.

Nuclei of Normal Rational Curves

Abstract

A -nucleus of a normal rational curve in PG is the intersection over all -dimensional osculating subspaces of the curve (). It is well known that for characteristic zero all nuclei are empty. In case of characteristic and the number of non-zero digits in the representation of in base equals the number of distinct nuclei. An explicit formula for the dimensions of -nuclei is given for .

Paper Structure

This paper contains 3 sections, 5 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. On Pascal's Triangle modulo $p$
  3. Nuclei

Key Result

LEMMA 1

(Lucas) Let $\langle n_\lambda\rangle$ and $\langle j_\lambda\rangle$ be the representations of non--negative integers $n$ and $j$ in base $p$. Then

Theorems & Definitions (9)

  • LEMMA 1
  • DEFINITION 1
  • DEFINITION 2
  • LEMMA 2
  • LEMMA 3
  • LEMMA 4
  • LEMMA 5
  • REMARK 1
  • DEFINITION 3