Veroneseans, power subspaces and independence

TL;DR

This paper investigates how the Veronese map and related power maps enhance the independence of points and subspaces, leading to the construction of large independent families in algebraic and geometric contexts.

Contribution

It demonstrates that the Veronese map often increases independence among points and subspaces, providing new methods to construct independent families.

Findings

d+1 Veronesean points of degree d are independent

dth power map increases independence of points and subspaces

Produces natural d+1-independent families of subspaces

Abstract

Results are proved indicating that the Veronese map v_d often increases independence of both sets of points and sets of subspaces. For example, any d+1 Veronesean points of degree d are independent. Similarly, the dth power map on the space of linear forms of a polynomial algebra also often increases independence of both sets of points and sets of subspaces. These ideas produce d+1-independent families of subspaces in a natural manner.