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The Stratification by Rank for Homogeneous Polynomials with Border Rank 5 which Essentially Depend on Five Variables

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Abstract

We give the stratification by the symmetric tensor rank of all degree d ≥ 9 homogeneous polynomials with border rank 5 and which depend essentially on at least five variables, extending previous works (A. Bernardi, A. Gimigliano, M. Idà, E. Ballico) on lower border ranks. For the polynomials which depend on at least five variables, only five ranks are possible: 5, d + 3, 2d + 1, 3d − 1, 4d − 3, but each of the ranks 3d − 1 and 2d + 1 is achieved in two geometrically different situations. These ranks are uniquely determined by a certain degree 5 zero-dimensional scheme A associated with the polynomial. The polynomial f depends essentially on at least five variables if and only if A is linearly independent (in all cases, f essentially depends on exactly five variables). The polynomial has rank 4d − 3 (resp. 3d − 1, resp. 2d + 1, resp. d + 3, resp. 5) if A has 1 (resp. 2, resp. 3, resp. 4, resp. 5) connected component. The assumption d ≥ 9 guarantees that each polynomial has a uniquely determined associated scheme A. In each case, we describe the dimension of the families of the polynomials with prescribed rank, each irreducible family being determined by the degrees of the connected components of the associated scheme A.

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Acknowledgments

We thank the referee for his useful comments. This study is partially supported by the MIUR and GNSAGA of INdAM.

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  1. Department of Mathematics, University of Trento, 38123, Trento, Italy

    Edoardo Ballico

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  1. Edoardo Ballico
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Correspondence to Edoardo Ballico.

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Ballico, E. The Stratification by Rank for Homogeneous Polynomials with Border Rank 5 which Essentially Depend on Five Variables. Acta Math Vietnam 42, 509–531 (2017). https://doi.org/10.1007/s40306-017-0211-6

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  • DOI: https://doi.org/10.1007/s40306-017-0211-6

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