Homogeneous Algebraic Complexity Theory and Algebraic Formulas

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

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Author Details

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. Homogeneous Algebraic Complexity Theory and Algebraic Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 43:1-43:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study algebraic complexity classes and their complete polynomials under homogeneous linear projections, not just under the usual affine linear projections that were originally introduced by Valiant in 1979. These reductions are weaker yet more natural from a geometric complexity theory (GCT) standpoint, because the corresponding orbit closure formulations do not require the padding of polynomials. We give the first complete polynomials for VF, the class of sequences of polynomials that admit small algebraic formulas, under homogeneous linear projections: The sum of the entries of the non-commutative elementary symmetric polynomial in 3 by 3 matrices of homogeneous linear forms. Even simpler variants of the elementary symmetric polynomial are hard for the topological closure of a large subclass of VF: the sum of the entries of the non-commutative elementary symmetric polynomial in 2 by 2 matrices of homogeneous linear forms, and homogeneous variants of the continuant polynomial (Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of circuits with arity-3 product gates.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Homogeneous polynomials
  • Waring rank
  • Arithmetic formulas
  • Border complexity
  • Geometric Complexity theory
  • Symmetric polynomials


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