Linear subspaces of symmetric tensors whose non-zero elements have at least a prescribed rank

E Ballico

doi:10.1007/s40065-014-0099-0

Abstract

We introduce large vector spaces M of multivariate homogeneous polynomials with a prescribed lower bound for the rank of each non-zero element of M.

Highlights

We consider linear spaces, W, of symmetric tensors, but we do not claim that our examples may be used to give nice codes, because in our examples all symmetric tensors T ∈ W have rank ≥ δ even over the algebraic closure of the base field
The second input came from our previous work [4,5], in which certain vector spaces of homogeneous polynomials are a key tool (the projective spaces W (O1, . . . , Ok; d) defined below are the projectivations of the vector spaces we consider in this paper)
Taking a scheme W1 ⊇ W, we reduce to the case (Wred) = s + 1

Summary

Introduction

By the bivariate case ([9,15,22], Theorem 4.1) we know that brXm,d (P) + rXm,d (P) = d + 2 and that there is a zero-dimensional scheme Z ⊂ Li, j such that P ∈ νd (Z ) , P ∈/ νd (Z ) for any Z Z and deg(Z ) = brX1,d (P). Remark 2.1 gives the existence of a minimal zero-dimensional scheme W ⊂ Pm such that

Results

Conclusion