Nuclei of Normal Rational Curves

TL;DR

This paper investigates the properties and dimensions of nuclei of normal rational curves in projective spaces over fields with positive characteristic, providing explicit formulas and linking nuclei to base-p digit representations.

Contribution

It introduces explicit formulas for the dimensions of nuclei of normal rational curves in positive characteristic fields, connecting these to the base-p digit structure of n+1.

Findings

Nuclei are empty in characteristic zero.

Number of nuclei relates to the number of non-zero base-p digits of n+1.

Explicit formulas for nuclei dimensions are derived for fields with size at least k+1.

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 $# F\geq k+1$.