Abstract
Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree dge 3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers underline{{mathrm {VSP}}} of these forms in the corresponding multigraded Hilbert schemes.
Highlights
Notions of ranks abound in the literature, perhaps because of their natural appearance in the realms of algebra and geometry, and in numerous applications thereof; see [18,22] and references therein for an introduction to the subject
Forms with vanishing Hessian were originally studied by Hesse in two classical papers [16,17], where the author tried to prove that these homogeneous polynomials are necessarily not concise
We provide a possible extension of the results of Bläser and Lysikov [2] to d-way tensors: Theorem Let T be a symmetric d-way tensor of minimal border rank
Summary
Notions of ranks abound in the literature, perhaps because of their natural appearance in the realms of algebra and geometry, and in numerous applications thereof; see [18,22] and references therein for an introduction to the subject. Forms with vanishing Hessian were originally studied by Hesse in two classical papers [16,17], where the author tried to prove that these homogeneous polynomials are necessarily not concise (or, in more geometric terms, that the hypersurfaces they define are cones) Thereafter, in their important work [12], Gordan and Noether showed that Hesse’s claim is true in the regime of at most four variables, whereas there exist infinitely many counterexamples afterwords. Theorem There exist concise minimal border rank wild forms of any degree d ≥ 3. We show that when n ≥ 10, their VSP’s are reducible; see Theorem 7.9
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