A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fröberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree 7 and in degree 9.

On the strength of general polynomials / Bik, A.; Oneto, A.. - In: LINEAR & MULTILINEAR ALGEBRA. - ISSN 0308-1087. - 2021:(2021), pp. 1-27. [10.1080/03081087.2021.1947955]

On the strength of general polynomials

Oneto A.
2021

Abstract

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fröberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree 7 and in degree 9.
Bik, A.; Oneto, A.
On the strength of general polynomials / Bik, A.; Oneto, A.. - In: LINEAR & MULTILINEAR ALGEBRA. - ISSN 0308-1087. - 2021:(2021), pp. 1-27. [10.1080/03081087.2021.1947955]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/326133
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