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Homogeneous Algebraic Complexity Theory and Algebraic Formulas

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

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  • 23 pages

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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

Cite AsGet BibTex

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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  1. M. Ben-Or and R. Cleve. Computing algebraic formulas using a constant number of registers. SIAM J. Comput., 21(21):54-58, 1992. URL:
  2. D. Bini. Relations between exact and approximate bilinear algorithms. Applications. Calcolo, 17(1):87-97, 1980. URL:
  3. D. Bini, M. Capovani, G. Lotti, and F. Romani. O(n^2.7799) complexity for n×n approximate matrix multiplication. Inform. Process. Lett., 8(5):234-235, 1979. URL:
  4. R. P. Brent. The Parallel Evaluation of General Arithmetic Expressions. J. Assoc. Comput. Mach., 21(2):201-206, 1974. URL:
  5. K. Bringmann, C. Ikenmeyer, and J. Zuiddam. On Algebraic Branching Programs of Small Width. J. ACM, 65(5):32:1-32:29, 2018. URL:
  6. W. Buczyńska and J. Buczyński. Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes. J. Alg. Geom., 23(1):63-90, 2014. Google Scholar
  7. W. Buczyńska and J. Buczyński. Apolarity, border rank, and multigraded Hilbert scheme. Duke Math. J., 170(16):3659-3702, 2021. URL:
  8. P. Bürgisser. Completeness and Reduction in Algebraic Complexity Theory, volume 7 of Algorithms and Computation in Mathematics. Springer Verlag, 2000. Google Scholar
  9. P. Bürgisser. The Complexity of Factors of Multivariate Polynomials. Found. Comp. Math., 4(4):369-396, 2004. URL:
  10. P. Bürgisser, M. Clausen, and M. A. Shokrollahi. Algebraic complexity theory, volume 315 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997. Google Scholar
  11. P. Bürgisser, C. Ikenmeyer, and G. Panova. No occurrence obstructions in geometric complexity theory. J. Amer. Math. Soc., 32(1):163-193, 2019. URL:
  12. P. Bürgisser, J. M. Landsberg, L. Manivel, and J. Weyman. An overview of mathematical issues arising in the Geometric Complexity Theory approach to VP ≠ VNP. SIAM J. Comput., 40(4):1179-1209, 2011. URL:
  13. A. Cayley. On the theory of linear transformations. Cambridge Math. J., iv:193-209, 1845. Google Scholar
  14. A. Clebsch. Zur Theorie der algebraischen Flächen. J. Reine Angew. Math., 58:93-108, 1861. Google Scholar
  15. P. Dutta, F. Gesmundo, C. Ikenmeyer, G. Jindal, and V. Lysikov. De-bordering and Geometric Complexity Theory for Waring rank and related models. arXiv:2211.07055, 2022. Google Scholar
  16. W. Fulton and J. Harris. Representation theory: a first course, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Google Scholar
  17. A. Iarrobino and V. Kanev. Power sums, Gorenstein algebras, and determinantal loci, volume 1721 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. URL:
  18. C. Ikenmeyer. Geometric complexity theory, tensor rank, and Littlewood-Richardson coefficients. PhD thesis, Universität Paderborn, 2013. Google Scholar
  19. C. Ikenmeyer and G. Panova. Rectangular Kronecker coefficients and plethysms in geometric complexity theory. Adv. Math., 319:40-66, 2017. URL:
  20. C. Ikenmeyer and A. Sanyal. A note on VNP-completeness and border complexity. Information Processing Letters, 176:106243, 2022. Google Scholar
  21. J. Jelisiejew. Pathologies on the Hilbert scheme of points. Inventiones mathematicae, 220(2):581-610, 2020. Google Scholar
  22. J. Jelisiejew and T. Mańdziuk. Limits of saturated ideals. arXiv, pages 1-31, 2022. URL:
  23. H. Kadish and J. M. Landsberg. Padded polynomials, their cousins, and geometric complexity theory. Comm. Algebra, 42(5):2171-2180, 2014. URL:
  24. H. Kraft. Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. Google Scholar
  25. M. Kumar. On the power of border of depth-3 arithmetic circuits. ACM Trans. Comput. Theory, 12(1):5:1-5:8, 2020. URL:
  26. D. Medini and A. Shpilka. Hitting Sets and Reconstruction for Dense Orbits in VP_e and ΣΠΣ Circuits. 36th Computational Complexity Conference (CCC 2021), 200:19:1-19:27, 2021. URL:
  27. K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput., 31(2):496-526, 2001. URL:
  28. K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory II: Towards explicit obstructions for embeddings among class varieties. SIAM J. Computing, 38(3):1175-1206, 2008. Google Scholar
  29. F. Palatini. Sulle superficie algebriche i cui S_h (h+1)-seganti non riempiono lo spazio ambiente. Atti della R. Acc. delle Scienze di Torino, 41:634-640, 1906. Google Scholar
  30. R. Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github Survey, 2021. URL:
  31. J. J. Sylvester. On the principles of the calculus of forms. J. Cambridge and Dublin Math., 7:52-97, 1852. Google Scholar
  32. A. Terracini. Sulle v_k per cui la varietà degli s_h(h+1)-seganti ha dimensione minore dell'ordinario. Rend. Circ. Mat., 31:392-396, 1911. Google Scholar
  33. S. Toda. Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Transactions on Information and Systems, 75(1):116-124, 1992. Google Scholar
  34. L. G. Valiant. Completeness classes in algebra. In Proceedings of the 11h Annual ACM Symposium on Theory of Computing, pages 249-261, 1979. URL:
  35. L. G. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff. Fast parallel computation of polynomials using few processors. SIAM J. Comput., 12(4):641-644, 1983. URL:
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