Fixed-Parameter Debordering of Waring Rank

Authors Pranjal Dutta , Fulvio Gesmundo , Christian Ikenmeyer , Gorav Jindal , Vladimir Lysikov

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Pranjal Dutta
  • School of Computing, National University of Singapore (NUS), Singapore
Fulvio Gesmundo
  • Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France
Christian Ikenmeyer
  • University of Warwick, UK
Gorav Jindal
  • Max Planck Institute for Software Systems, Saarbrücken, Germany
Vladimir Lysikov
  • Ruhr-Universität Bochum, Germany

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Pranjal Dutta, Fulvio Gesmundo, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov. Fixed-Parameter Debordering of Waring Rank. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • border complexity
  • Waring rank
  • debordering
  • apolarity


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