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Shao's polynomial bound conjecture for the primitive degree of tensors

Let A\mathbb{A} be a nonnegative primitive tensor of fixed order mm and dimension nn. Its primitive degree γ(A)\gamma(\mathbb{A}) is the smallest positive integer rr such that Ar\mathbb{A}^r is essentially positive, equivalently such that the majorization matrix M(Ar)M(\mathbb{A}^r) is positive, where M(A)ij=aij…jM(\mathbb{A})_{ij}=a_{ij\ldots j}. Shao's conjecture. When mm is fixed, there exists a polynomial f(n)f(n) such that

γ(A)≤f(n)\gamma(\mathbb{A})\leq f(n)

for every nonnegative primitive tensor A\mathbb{A} of order mm and dimension nn. The conjecture asks for a polynomial upper bound, in the dimension, on the primitive degree of tensors of each fixed order; the supplied text gives no resolution, so its status remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Shao's polynomial bound conjecture for the primitive degree of tensors

    Let A\mathbb{A} be a nonnegative primitive tensor of fixed order mm and dimension nn, and let γ(A)\gamma(\mathbb{A}) denote its primitive degree. Shao's conjecture. When mm is fixed, there exists a polynomial f(n)f(n) in nn such that

    γ(A)≤f(n)\gamma(\mathbb{A})\leq f(n)

    for all nonnegative primitive tensors of order mm and dimension nn. The conjecture asks for a polynomial upper bound on the primitive degree when the tensor order is fixed; the supplied source does not establish its resolution.

    source: Pingzhi Yuany, Zilong He and Lihua You, “A conjecture on the primitive degree of Tensors”, arXiv:1310.8461 (2013).

Sources & referencesView supporting material

Primary source

Lihua You, Yafei Chen and Pingzhi Yuan, “Some results of strongly primitive tensors”, arXiv:1705.04554 (2017).

Additional references

3 papers in this index state this conjecture (2013–2017). The statement above is taken from the most recent of them; the others are arXiv:1408.3457, arXiv:1310.8461.

combinatoricsnonnegative-tensorstensor-theory

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