2017/02/07 by Edoardo Ballico, Ballico, Edoardo
Computer Science · Mathematics · #14N05 #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Matrix Theory and Algorithms #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1702.01914
openalex publication_date 2017/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 5 variables, extending previous works (A. Bernardi, A. Gimigliano, M. Idà, E. Ballico) on lower border ranks. For the polynomials which depend on at least 5 variables only 5 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 to the polynomial. The polynomial f depends essentially on at least 5 variables if and only if A is linearly independent (in all cases f essentially depends on exactly 5 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 components. 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.