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

Abstract

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

Veronese Varieties over Fields with non-zero Characteristic: A Survey

“…at the given point is the k-dimensional projective subspace spanned by the first k + 1 columns of the matrix (10). The derivative points at…”

“…In order to find all invariant subspaces, we follow J. Gmainer [10]: Suppose that the dimension n is fixed. For j ∈ N let…”

“…Theorem 4 [10] Let #F ≥ n+ 2 or n = 2. A subspace U is invariant under the collineation group of the normal rational curve (9) if, and only if, U is spanned by base points F c λ with λ ∈ Λ ⊂ {0, 1, .…”

“…Theorem 6 [10] Let the positions of the non-zero digits of b := n + 1 in base p be denoted by N 1 , N 2 , . .…”

A Topological View of Reed–Solomon Codes

“…Let us call V t n the Veronese image under the Veronese mapping given by: The Veronese image of each r-dimensional subspace of PG(n, p) is a sub-Veronesean variety V t r of V t n , and all those subspaces are indexed by partitions in P (t). Thus by a Theorem due to Gmainer are invariant under the collineation group of the normal rational curve (see [16]).…”