On the rank of a symmetric form

By Kristian Ranestad, Frank-Olaf Schreyer. 2011

87 citations recorded by the source corpus.

We give a lower bound for the degree of a finite apolar subscheme of a symmetric form F, in terms of the degrees of the generators of the annihilator ideal of F. In the special case, when F is a monomial x_0^d_0 x_2^d_2... x_n^d_n with d_0<= d_1<=...<=d_n-1<= d_n we deduce that the minimal length of an apolar subscheme of F is (d_0+1)...(d_n-1+1), and if d_0=..=d_n, then this minimal length coincides with the rank of F.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bfe8cdfa2847757758a0509a56253b5a80cbc1d0

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Waring Decompositions of Monomials (2012, methodology): “0 ···x dn n is a monomial. We answer the question of uniqueness of Waring decomposition up to scaling the variables, which amounts to determining whether a torus action on VSP◦(F) is transitive. Both [RS11] and [CCG12] noted that a Waring decomposition F= ℓd 1 + ···+ ℓd r may be obtained with {[ℓ1],”
A comparison of different notions of ranks of symmetric tensors (2012, background): “This leads to the notion of catalecticant rank r H ( f ) of a tensor f , which is also called “diﬀerential length” in [26][Deﬁnition 5.66, p.198].”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “80) to obtain lower bounds to the symmetric-rank of a homogeneous polynomial was first given in [170].”