Apolarity and direct sum decomposability of polynomials

2013/07/12 by Buczyńska, Weronika, Buczyński, Jarosław, Kleppe, Johannes +1 · 3 citations
#13H10 #14N15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.3314

Abstract

A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct sum, in terms of the apolar ideal of the polynomial. We prove that the apolar ideal of a polynomial of degree d strictly depending on all variables has a minimal generator of degree d if and only if it is a limit of direct sums.

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• On strongly Koszul algebras and tidy Gröbner bases
• Components of Hilbert Schemes of low degree and smoothable algebras
• Detecting Direct Sums of Tensors and Their Limits

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