Bounded-rank tensors are defined in bounded degree

By Jan Draisma, J. Kuttler. 2011

70 citations recorded by the source corpus.

Matrices of rank at most k are defined by the vanishing of polynomials of degree k+1 in their entries (namely, their ((k+1)×(k+1))-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d=d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its size in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial ways.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/04076b23bba0bdf4f3df519641e8c5d30d4ee57e

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In Computational Mathematics

1. Jan Draisma
2. Walter Gubler
3. Eric Katz

In Computational Theory and Mathematics

1. Jan Draisma
2. Walter Gubler
3. Eric Katz

In Numerical Analysis

1. Jan Draisma

Citations in the corpus

• Topological Noetherianity of polynomial functors (2017, methodology): “The idea further to do induction on δX and to use directional derivatives is new, but inspired by techniques used earlier in [10, 12, 11], where a determinant is regarded as an affine-linear polynomial in one matrix entry, whose coefficient is a determinant of lower order, and induction is done over”
• Noetherianity up to Symmetry (2013, unknown): “The set-theoretic theorem above was first proved in [ DK13 ] for k Seg , i.e., for any fixed secant variety of Seg ; and a discussion with Snowden led to the insight that our proof generalises to bounded, good, ∆-varieties as in the theorem.”