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

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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

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