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On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares

Authors Ilario Bonacina, Nicola Galesi, Massimo Lauria

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Author Details

Ilario Bonacina
  • Universitat Politècnica de Catalunya, Barcelona, Spain
Nicola Galesi
  • Sapienza Università di Roma, Italy
Massimo Lauria
  • Sapienza Università di Roma, Italy


The authors would like to thank Albert Atserias for fruitful discussions.

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Ilario Bonacina, Nicola Galesi, and Massimo Lauria. On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Computing methodologies → Representation of polynomials
  • polynomial calculus
  • sum-of-squares
  • roots of unity
  • knapsack


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  1. Albert Atserias and Tuomas Hakoniemi. Size-degree trade-offs for sums-of-squares and positivstellensatz proofs. In Amir Shpilka, editor, 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019, New Brunswick, NJ, USA, volume 137 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  2. Albert Atserias and Joanna Ochremiak. Proof complexity meets algebra. ACM Trans. Comput. Logic, 20(1), December 2018. Google Scholar
  3. Roberto J Bayardo Jr and Robert Schrag. Using CSP look-back techniques to solve real-world SAT instances. In AAAI/IAAI, pages 203-208, 1997. Google Scholar
  4. Christoph Berkholz. The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), volume 96 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:14, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  5. Grigoriy Blekherman, João Gouveia, and James Pfeiffer. Sums of squares on the hypercube. Mathematische Zeitschrift, pages 1-14, 2016. URL:
  6. Grigoriy Blekherman and Cordian Riener. Symmetric non-negative forms and sums of squares. Discrete and Computational Geometry, 65(3):764-799, May 2020. URL:
  7. Samuel R. Buss, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. Linear gaps between degrees for the polynomial calculus modulo distinct primes. J. Comput. Syst. Sci., 62(2):267-289, 2001. URL:
  8. Matthew Clegg, Jeff Edmonds, and Russell Impagliazzo. Using the Gröbner basis algorithm to find proofs of unsatisfiability. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 174-183. ACM, 1996. Google Scholar
  9. John Conway and A. Jones. Trigonometric diophantine equations (on vanishing sums of roots of unity). Acta Arithmetica, 30(3):229-240, 1976. URL:
  10. David Cox, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms : An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edition. Springer, 2007. Google Scholar
  11. Jesús A De Loera, J. Lee, S. Margulies, and S. Onn. Expressing combinatorial problems by systems of polynomial equations and Hilbert’s Nullstellensatz. Comb. Probab. Comput., 18(4):551-582, July 2009. URL:
  12. Jesús A De Loera, Jon Lee, Peter N Malkin, and Susan Margulies. Computing infeasibility certificates for combinatorial problems through Hilbert’s Nullstellensatz. Journal of Symbolic Computation, 46(11):1260-1283, 2011. Google Scholar
  13. Jesús A De Loera, Susan Margulies, Michael Pernpeintner, Eric Riedl, David Rolnick, Gwen Spencer, Despina Stasi, and Jon Swenson. Graph-coloring ideals: Nullstellensatz certificates, Gröbner bases for chordal graphs, and hardness of Gröbner bases. In Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, pages 133-140. ACM, 2015. Google Scholar
  14. Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, and Dmitry Sokolov. The Power of Negative Reasoning. In 36th Computational Complexity Conference (CCC 2021), volume 200 of Leibniz International Proceedings in Informatics (LIPIcs), pages 40:1-40:24, 2021. URL:
  15. R. Dvornicich and U. Zannier. Sums of roots of unity vanishing modulo a prime. Archiv der Mathematik, 79(2):104-108, August 2002. URL:
  16. Roberto Dvornicich and Umberto Zannier. On sums of roots of unity. Monatshefte für Mathematik, 129(2):97-108, February 2000. URL:
  17. D. Grigoriev. Complexity of positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, December 2001. Google Scholar
  18. Dima Grigoriev. Tseitin’s tautologies and lower bounds for Nullstellensatz proofs. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 648-652. IEEE Computer Society, 1998. URL:
  19. R. Impagliazzo, P. Pudlák, and J. Sgall. Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Computational Complexity, 8(2):127-144, November 1999. Google Scholar
  20. Daniela Kaufmann, Paul Beame, Armin Biere, and Jakob Nordström. Adding dual variables to algebraic reasoning for gate-level multiplier verification. In Proceedings of the 25th Design, Automation and Test in Europe Conference (DATE'22), 2022. Google Scholar
  21. Daniela Kaufmann and Armin Biere. Nullstellensatz-proofs for multiplier verification. In Computer Algebra in Scientific Computing - 22nd International Workshop, CASC 2020, Linz, Austria, September 14-18, 2020, Proceedings, pages 368-389, 2020. URL:
  22. Daniela Kaufmann, Armin Biere, and Manuel Kauers. Verifying large multipliers by combining SAT and computer algebra. In 2019 Formal Methods in Computer Aided Design, FMCAD 2019, San Jose, CA, USA, October 22-25, 2019, pages 28-36, 2019. URL:
  23. Daniela Kaufmann, Armin Biere, and Manuel Kauers. From DRUP to PAC and back. In 2020 Design, Automation & Test in Europe Conference & Exhibition, DATE 2020, Grenoble, France, March 9-13, 2020, pages 654-657, 2020. URL:
  24. T.Y Lam and K.H Leung. On vanishing sums of roots of unity. Journal of Algebra, 224(1):91-109, 2000. Google Scholar
  25. J. Lasserre. An explicit exact SDP relaxation for nonlinear 0-1 programs. Integer Programming and Combinatorial Optimization, pages 293-303, 2001. Google Scholar
  26. Massimo Lauria and Jakob Nordström. Graph Colouring is Hard for Algorithms Based on Hilbert’s Nullstellensatz and Gröbner Bases. In 32nd Computational Complexity Conference (CCC 2017), volume 79, pages 2:1-2:20, 2017. URL:
  27. Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the sum-of-squares degree of symmetric quadratic functions. In 31st Conference on Computational Complexity, volume 50 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 17, 31. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016. Google Scholar
  28. João P. Marques-Silva and Karem A. Sakallah. GRASP: A search algorithm for propositional satisfiability. Computers, IEEE Transactions on, 48(5):506-521, 1999. Google Scholar
  29. M.W. Moskewicz, C.F. Madigan, Y. Zhao, L. Zhang, and S. Malik. Chaff: Engineering an efficient SAT solver. In Proceedings of the 38th annual Design Automation Conference, pages 530-535. ACM, 2001. Google Scholar
  30. Pablo A. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical programming, 96(2):293-320, 2003. Google Scholar
  31. Aaron Potechin. Sum of Squares Bounds for the Ordering Principle. In Shubhangi Saraf, editor, 35th Computational Complexity Conference (CCC 2020), volume 169 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:37, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL:
  32. Grant Schoenebeck. Linear level Lasserre lower bounds for certain k-CSPs. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 593-602. IEEE Computer Society, 2008. URL:
  33. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 77-82. ACM, 1987. URL:
  34. Dmitry Sokolov. (Semi)Algebraic proofs over ± 1 variables. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing. ACM, June 2020. Google Scholar
  35. Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 303-312. ACM, 2009. URL:
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