Pascal’s Triangle, Normal Rational Curves, and their Invariant Subspaces

Pascal's Triangle, Normal Rational Curves, and their Invariant Subspaces

Each normal rational curve Γ in PG(n, F ) admits a group PΓL(Γ) of automorphic collineations. It is well known that for characteristic zero only the empty and the entire subspace are PΓL(Γ)–invariant. In case of characteristic p > 0 there may be further invariant subspaces. For #F ≥ n+ 2, we give a construction of all PΓL(Γ)–invariant subspaces. It turns out that the corresponding lattice is to...

Invariant curves and semiconjugacies of rational functions

Jordan analytic curves which are invariant under rational functions are studied.

On Invariant Subspaces of Normal Operators1

Nuclei of Normal Rational Curves

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 ∈ {−1, 0, . . . , n− 1}). It is well known that for characteristic zero all nuclei are empty. In case of characteristic p > 0 and #F ≥ 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 ...

Completeness of Normal Rational Curves

The completeness of normal rational curves, considered as (q + 1)-arcs in PG(n, q), is investigated. Previous results of Storme and Thas are improved by using a result by Kovacs. This solves the problem completely for large prime numbers q and odd nonsquare prime powers q = p2h+l with p prime, p > po(h), h>\,1where po(h) is an odd prime number which depends on h.