computational complexity

, Volume 26, Issue 4, pp 881–909 | Cite as

Sparse multivariate polynomial interpolation on the basis of Schubert polynomials



Schubert polynomials were discovered by A. Lascoux and M. Schützenberger in the study of cohomology rings of flag manifolds in 1980s. These polynomials generalize Schur polynomials and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials and expand in the Schubert basis with the generalized Littlewood–Richardson coefficients. In this paper, we initiate the study of these two families of polynomials from the perspective of computational complexity theory. We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in #P (when evaluating on nonnegative integral inputs) and VNP. Our main result is a deterministic algorithm that computes the expansion of a polynomial f of degree d in \({\mathbb{Z}[x_1,\dots, x_n]}\) on the basis of Schubert polynomials, assuming an oracle computing Schubert polynomials. This algorithm runs in time polynomial in n, d, and the bit size of the expansion. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Adv Appl Math 18(3):271–285, 1997). In fact, our interpolation algorithm is general enough to accommodate any linear basis satisfying certain natural properties. Applications of the above results include a new algorithm that computes the generalized Littlewood–Richardson coefficients.


Schubert polynomials Sparse interpolation #P VNP 

Subject classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Vikraman Arvind, Partha Mukhopadhyay, Srinivasan Srikanth (2010) New Results on Noncommutative and Commutative Polynomial Identity Testing. Computational Complexity 19(4): 521–558CrossRefMATHMathSciNetGoogle Scholar
  2. Sami Assaf, Nantel Bergeron & Frank Sottile (2014). A combinatorial proof that Schubert vs. Schur coefficients are nonnegative. arXiv preprint arXiv:1405.2603.
  3. Alexander Barvinok, Fomin Sergey (1997) Sparse interpolation of symmetric polynomials. Advances in Applied Mathematics 18(3): 271–285CrossRefMATHMathSciNetGoogle Scholar
  4. Michael Ben-Or & Prasoon Tiwari (1988). A Deterministic Algorithm for Sparse Multivariate Polynominal Interpolation (Extended Abstract). In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2–4, 1988, Chicago, Illinois, USA, 301–309.URL
  5. Nantel Bergeron, Billey Sara (1993) RC-graphs and Schubert polynomials. Experimental Mathematics 2(4): 257–269CrossRefMATHMathSciNetGoogle Scholar
  6. Nader H. Bshouty & Richard Cleve (1998). Interpolating Arithmetic Read-Once Formulas in Parallel. SIAM J. Comput. 27(2), 401–413.Google Scholar
  7. Peter Bürgisser (2000). Completeness and reduction in algebraic complexity theory, volume 7. Springer Science & Business Media.Google Scholar
  8. Peter Bürgisser & Christian Ikenmeyer (2013). Deciding Positivity of Littlewood-Richardson Coefficients. SIAM J. Discrete Math. 27(4), 1639–1681 URL
  9. Cassio P de Campos, Georgios Stamoulis & Dennis Weyland (2013). A Structured View on Weighted Counting with Relations to Quantum Computation and Applications. In Electronic Colloquium on Computational Complexity (ECCC), volume 20, 133.Google Scholar
  10. Ankit Gupta, Neeraj Kayal & Satyanarayana V. Lokam (2011). Efficient Reconstruction of Random Multilinear Formulas. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22–25, 2011, 778–787. URL
  11. Ankit Gupta, Neeraj Kayal & Satyanarayana V. Lokam (2012). Reconstruction of depth-4 multilinear circuits with top fan-in 2. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19–22, 2012, 625–642. URL
  12. Ankit Gupta, Neeraj Kayal & Youming Qiao (2014). Random arithmetic formulas can be reconstructed efficiently. Computational Complexity 23(2), 207–303. URL
  13. Erich Kaltofen & Yagati N. Lakshman (1988). Improved Sparse Multivariate Polynomial Interpolation Algorithms. In Symbolic and Algebraic Computation, International Symposium ISSAC’88, Rome, Italy, July 4–8, 1988, Proceedings, Patrizia M. Gianni, editor, volume 358 of Lecture Notes in Computer Science, 467–474. Springer. ISBN 3-540-51084-2. URL
  14. Zohar Shay Karnin & Amir Shpilka (2009). Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15–18 July 2009, 274–285. URL
  15. Neeraj Kayal (2012). Affine projections of polynomials: extended abstract. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19–22, 2012, 643–662. URL
  16. Adalbert Kerber, Axel Kohnert & Alain Lascoux (1992). Symbolic Computation in Combinatorics SYMMETRICA, an object oriented computer-algebra system for the symmetric group. Journal of Symbolic Computation 14(2), 195 – 203. ISSN 0747-7171. URL
  17. Adam Klivans & Daniel A. Spielman (2001). Randomness efficient identity testing of multivariate polynomials. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6–8, 2001, Heraklion, Crete, Greece, 216–223. URL
  18. Allen Knutson & Ezra Miller (2005). Gröbner Geometry of Schubert Polynomials. Annals of Mathematics 161(3), pp. 1245–1318. ISSN 0003486X. URL
  19. Kogan Mikhail (2001) RC-graphs and a generalized Littlewood-Richardson rule. International Mathematics Research Notices 2001(15): 765–782CrossRefMATHMathSciNetGoogle Scholar
  20. Pascal Koiran (2005). Valiant’s model and the cost of computing integers. Computational Complexity 13(3–4), 131–146. URL
  21. Alain Lascoux (2003). Symmetric functions and combinatorial operators on polynomials, volume 99. American Mathematical Soc.Google Scholar
  22. Alain Lascoux (2008). Schubert and Macdonald polynomials, a parallel. Electronically available at
  23. Alain Lascoux (2013). Polynomials. Electronically available at
  24. Alain Lascoux & Marcel-Paul Schützenberger (1982). Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294(13), 447–450Google Scholar
  25. Alain Lascoux & Marcel-Paul Schützenberger (1985). Schubert polynomials and the Littlewood-Richardson rule. Letters in Mathematical Physics 10(2–3), 111–124.Google Scholar
  26. Cristian Lenart, Sottile Frank (2003) Skew Schubert polynomials. Proceedings of the American Mathematical Society 131(11): 3319–3328CrossRefMATHMathSciNetGoogle Scholar
  27. Ian Grant Macdonald (1991). Notes on Schubert polynomials, volume 6. Montréal: Dép. de mathématique et d’informatique, Université du Québec à Montréal.Google Scholar
  28. L. Manivel (2001). Symmetric Functions, Schubert Polynomials, and Degeneracy Loci. Collection SMF.: Cours spécialisés. American Mathematical Society. ISBN 9780821821541. URL
  29. Karola Mészáros, Greta Panova & Alexander Postnikov (2014). Schur Times Schubert via the Fomin-Kirillov Algebra. Electr. J. Comb. 21(1), P1.39. URL
  30. Ketan Mulmuley (2011). On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna. J. ACM 58(2), 5. URL
  31. Ketan D Mulmuley, Hariharan Narayanan & Milind Sohoni (2012). Geometric complexity theory III: on deciding nonvanishing of Littlewood-Richardson coefficient. Journal of Algebraic Combinatorics 36(1), 103–110.Google Scholar
  32. Hariharan Narayanan (2006) On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients. Journal of Algebraic Combinatorics 24(3): 347–354CrossRefMATHMathSciNetGoogle Scholar
  33. R. Saptharishi (2016). A survey of lower bounds in arithmetic circuit complexity. URL Version 3.0.0.
  34. Amir Shpilka (2009) Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates. SIAM J. Comput. 38(6): 2130–2161CrossRefMATHMathSciNetGoogle Scholar
  35. Amir Shpilka & Ilya Volkovich (2015). Read-once polynomial identity testing. Computational Complexity 24(3), 477–532. URL
  36. Amir Shpilka & Amir Yehudayoff (2010). Arithmetic Circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science 5, 207–388. ISSN 1551-305X.
  37. Leslie G. Valiant (1979). Completeness Classes in Algebra. In Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard & Alfred V. Aho, editors, 249–261. ACM. URL
  38. Richard Zippel (1979). Probabilistic algorithms for sparse polynomials. In Symbolic and Algebraic Computation, EdwardW. Ng, editor, volume 72 of Lecture Notes in Computer Science, 216–226. Springer Berlin Heidelberg. ISBN 978-3-540-09519-4. URL
  39. Richard Zippel (1990) Interpolating Polynomials from Their Values. J. Symb. Comput. 9(3): 375–403CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  2. 2.Centre for Quantum Computation and Intelligent SystemsUniversity of Technology SydneySydneyAustralia

Personalised recommendations