Testing Polynomial Equivalence by Scaling Matrices

  • Markus BläserEmail author
  • B. V. Raghavendra Rao
  • Jayalal Sarma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


In this paper we study the polynomial equivalence problem: test if two given polynomials f and g are equivalent under a non-singular linear transformation of variables.

We begin by showing that the more general problem of testing whether f can be obtained from g by an arbitrary (not necessarily invertible) linear transformation of the variables is equivalent to the existential theory over the reals. This strengthens an \(\mathsf {NP}\)-hardness result by Kayal [9].

Two n-variate polynomials f and g are said to be equivalent up to scaling if there are scalars \(a_1, \ldots , a_n \in \mathbb {F} \setminus \{0\}\) such that \(f(a_1x_1,\ldots , a_nx_n) = g(x_1,\ldots , x_n)\). Testing whether two polynomials are equivalent by scaling matrices is a special case of the polynomial equivalence problem and is harder than the polynomial identity testing problem.

As our main result, we obtain a randomized polynomial time algorithm for testing if two polynomials are equivalent up to a scaling of variables with black-box access to polynomials f and g over the real numbers.

An essential ingredient to our algorithm is a randomized polynomial time algorithm that given a polynomial as a black box obtains coefficients and degree vectors of a maximal set of monomials whose degree vectors are linearly independent. This algorithm might be of independent interest. It also works over finite fields, provided their size is large enough to perform polynomial interpolation.



The work was supported by the Indo-German Max Planck Center for Computer Science.


  1. 1.
    Agrawal, M., Saxena, N.: Equivalence of \(\mathbb{F}\)-algebras and cubic forms. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 115–126. Springer, Heidelberg (2006). doi: 10.1007/11672142_8 CrossRefGoogle Scholar
  2. 2.
    Bläser, M.: Explicit tensors. In: Agrawal, M., Arvind, V. (eds.) Perspectives in Computational Complexity. PCSAL, vol. 26, pp. 117–130. Springer, Cham (2014). doi: 10.1007/978-3-319-05446-9_6 Google Scholar
  3. 3.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften, vol. 315. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  4. 4.
    Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 460–467 (1988)Google Scholar
  5. 5.
    Dvir, Z., Oliveira, R.M., Shpilka, A.: Testing equivalence of polynomials under shifts. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 417–428. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43948-7_35 Google Scholar
  6. 6.
    Grigoriev, D.: Testing shift-equivalence of polynomials by deterministic, probabilistic and quantum machines. Theor. Comput. Sci. 180(1), 217–228 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13(1/2), 1–46 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kayal, N.: Efficient algorithms for some special cases of the polynomial equivalence problem. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, 23–25 January 2011, pp. 1409–1421 (2011)Google Scholar
  9. 9.
    Kayal, N.: Affine projections of polynomials: extended abstract. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, 19–22 May 2012, pp. 643–662 (2012)Google Scholar
  10. 10.
    Kayal, N., Nair, V., Saha, C., Tavenas, S.: Reconstruction of full rank algebraic branching programs. In: 32nd IEEE Conference on Computational Complexity (CCC) (2017, to appear)Google Scholar
  11. 11.
    Klivans, A.R., Spielman, D.A.: Randomness efficient identity testing of multivariate polynomials. In: Proceedings on 33rd Annual ACM Symposium on Theory of Computing, 6–8 July 2001, Heraklion, Crete, Greece, pp. 216–223 (2001)Google Scholar
  12. 12.
    Lang, S.: Algebra. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Patarin, J.: Hidden fields equations (HFE) and Isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996). doi: 10.1007/3-540-68339-9_4 CrossRefGoogle Scholar
  14. 14.
    Saxena, N.: Morphisms of rings and applications to complexity. Ph.D. thesis, Department of Computer Science, Indian Institute of Technology, Kanpur, India (2006)Google Scholar
  15. 15.
    Schaefer, M., Stefankovic, D.: The complexity of tensor rank. CoRR, abs/1612.04338 (2016)Google Scholar
  16. 16.
    Shitov, Y.: How hard is tensor rank? CoRR, abs/1611.01559 (2016)Google Scholar
  17. 17.
    Thierauf, T.: The isomorphism problem for read-once branching programs and arithmetic circuits. Chic. J. Theor. Comput. Sci. 1998(1) (1998)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Markus Bläser
    • 1
    Email author
  • B. V. Raghavendra Rao
    • 2
  • Jayalal Sarma
    • 2
  1. 1.Saarland Informatics CampusSaarland UniversitySaarbrückenGermany
  2. 2.IIT MadrasChennaiIndia

Personalised recommendations