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DOI: 10.4230/LIPIcs.FSTTCS.2022.20

Degree-Restricted Strength Decompositions and Algebraic Branching Programs

Authors: Fulvio Gesmundo, Purnata Ghosal, Christian Ikenmeyer, and Vladimir Lysikov

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract

We analyze Kumar’s recent quadratic algebraic branching program size lower bound proof method (CCC 2017) for the power sum polynomial. We present a refinement of this method that gives better bounds in some cases. The lower bound relies on Noether-Lefschetz type conditions on the hypersurface defined by the homogeneous polynomial. In the explicit example that we provide, the lower bound is proved resorting to classical intersection theory. Furthermore, we use similar methods to improve the known lower bound methods for slice rank of polynomials. We consider a sequence of polynomials that have been studied before by Shioda and show that for these polynomials the improved lower bound matches the known upper bound.

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Fulvio Gesmundo, Purnata Ghosal, Christian Ikenmeyer, and Vladimir Lysikov. Degree-Restricted Strength Decompositions and Algebraic Branching Programs. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gesmundo_et_al:LIPIcs.FSTTCS.2022.20,
  author =	{Gesmundo, Fulvio and Ghosal, Purnata and Ikenmeyer, Christian and Lysikov, Vladimir},
  title =	{{Degree-Restricted Strength Decompositions and Algebraic Branching Programs}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.20},
  URN =		{urn:nbn:de:0030-drops-174127},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.20},
  annote =	{Keywords: Lower bounds, Slice rank, Strength of polynomials, Algebraic branching programs}
}

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Degree-Restricted Strength Decompositions and Algebraic Branching Programs

Abstract

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