Fixed-parameter debordering of Waring rank
CoRR(2024)
摘要
Border complexity measures are defined via limits (or topological closures),
so that any function which can approximated arbitrarily closely by low
complexity functions itself has low border complexity. Debordering is the task
of proving an upper bound on some non-border complexity measure in terms of a
border complexity measure, thus getting rid of limits.
Debordering is at the heart of understanding the difference between Valiant's
determinant vs permanent conjecture, and Mulmuley and Sohoni's variation which
uses border determinantal complexity. The debordering of matrix multiplication
tensors by Bini played a pivotal role in the development of efficient matrix
multiplication algorithms. Consequently, debordering finds applications in both
establishing computational complexity lower bounds and facilitating algorithm
design. Currently, very few debordering results are known.
In this work, we study the question of debordering the border Waring rank of
polynomials. Waring and border Waring rank are very well studied measures in
the context of invariant theory, algebraic geometry, and matrix multiplication
algorithms. For the first time, we obtain a Waring rank upper bound that is
exponential in the border Waring rank and only linear in the degree. All
previous known results were exponential in the degree. For polynomials with
constant border Waring rank, our results imply an upper bound on the Waring
rank linear in degree, which previously was only known for polynomials with
border Waring rank at most 5.
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