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Register a new userPartially symmetric variants of Comons problem via simultaneous rank
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A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. By exploiting algebraic tools such as apolarity theory, we show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. This approach aims to understand to what extent the symmetries of a tensor affect its rank. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials.
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Given tensors $T$ and $T$ of order $k$ and $k$ respectively, the tensor product $T otimes T$ is a tensor of order $k+k$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry become available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called W-states, namely monomials of the form $x^{d-1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 -1}y otimes ... otimes x^{d_k-1} y$ is at most $2^{k-1}(d_1+ ... +d_k)$.
We show that the Waring rank of the $3 times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 times 3$ permanent is at least $14$.
We classify the subvarieties of infinite dimensional affine space that are stable under the infinite symmetric group. We determine the defining equations and point sets of these varieties as well as the containments between them.
We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique.
We prove that strength and slice rank of homogeneous polynomials of degree $d geq 5$ over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a conjecture of Catalisano, Geramita, Gimigliano, Harbourne, Migliore, Nagel and Shin concerning dimensions of secant varieties of the varieties of reducible homogeneous polynomials. These statements were already known in degrees $2leq dleq 7$ and $d=9$.
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