## Abstract

We introduce a novel take on sum-of-squares that is able to reason with complex numbers and still make use of polynomial inequalities. This proof system might be of independent interest since it allows to represent multivalued domains both with Boolean and Fourier encoding. We show degree and size lower bounds in this system for a natural generalization of knapsack: the vanishing sums of roots of unity. These lower bounds naturally apply to polynomial calculus as-well.

## Article PDF

### Similar content being viewed by others

## References

Yaroslav Alekseev, Dima Grigoriev, Edward A. Hirsch & Iddo Tzameret (2020). Semi-Algebraic Proofs, IPS Lower Bounds, and the τ -Conjecture: Can a Natural Number Be Negative?

*In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC’20)*, 54–67.Albert Atserias & Tuomas Hakoniemi (2019). Size-Degree Trade- Offs for Sums-of-Squares and Positivstellensatz Proofs.

*In Proceedings of the 34th Computational Complexity Conference (CCC’19)*, volume 137 of LIPIcs, 24:1–24:20.Albert Atserias & Joanna Ochremiak (2018). Proof Complexity Meets Algebra. ACM Trans. Comput. Logic 20(1).

Roberto J Bayardo Jr & Robert Schrag (1997). Using CSP look-back techniques to solve real-world SAT instances.

*In Proceedings of the 14th National Conference on Artificial Intelligence and 9th Conference on Innovative Applications of Artificial Intelligence*(AAAI’97/IAAI’97), 203–208.Christoph Berkholz (2018). The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs. In

*Proceedings of the 35th Symposium on Theoretical Aspects of Computer Science (STACS’18)*, volume 96, 11:1–11:14.Grigoriy Blekherman, João Gouveia & James Pfeiffer (2016). Sums of squares on the hypercube. Mathematische Zeitschrift 1–14.

Grigoriy Blekherman & Cordian Riener (2020). Symmetric Non- Negative Forms and Sums of Squares. Discrete and Computational Geometry 65(3), 764–799.

Ilario Bonacina, Nicola Galesi & Massimo Lauria (2022). On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum- Of-Squares. In

*Proceedings of the 47th International Symposium on Mathematical Foundations of Computer Science (MFCS’22)*, volume 241 of LIPIcs, 23:1–23:15.Samuel R. Buss, Dima Grigoriev, Russell Impagliazzo & Toniann Pitassi (2001). Linear Gaps between Degrees for the Polynomial Calculus Modulo Distinct Primes. J. Comput. Syst. Sci. 62(2), 267–289. https://doi.org/10.1006/jcss.2000.1726.

Matthew Clegg, Jeff Edmonds & Russell Impagliazzo (1996). Using the Gröbner Basis Algorithm to Find Proofs of Unsatisfiability. In

*Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC’96)*, 174–183.John Conway & A. Jones (1976). Trigonometric diophantine equations (On vanishing sums of roots of unity). Acta Arithmetica 30(3), 229–240.

David Cox, John Little & Donal O’Shea (2007). Ideals, Varieties, and Algorithms :

*An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edition*. Springer.Jesús A De Loera, J. Lee, S. Margulies & S. Onn (2009). Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert’s Nullstellensatz. Comb. Probab. Comput. 18(4), 551–582.

Jesús A De Loera, Jon Lee, Peter N Malkin & Susan Margulies (2011). Computing infeasibility certificates for combinatorial problems through Hilbert’s Nullstellensatz. Journal of Symbolic Computation 46(11), 1260–1283.

Jesús A De Loera, Susan Margulies, Michael Pernpeintner, Eric Riedl, David Rolnick, Gwen Spencer, Despina Stasi & Jon Swenson (2015). 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 (ISSAC’15), 133–140.

R. Dvornicich & U. Zannier (2002). Sums of roots of unity vanishing modulo a prime. Archiv der Mathematik 79(2), 104–108.

Roberto Dvornicich & Umberto Zannier (2000). On Sums of Roots of Unity. Monatshefte für Mathematik 129(2), 97–108.

Michael A. Forbes, Amir Shpilka, Iddo Tzameret & Avi Wigderson (2021). Proof Complexity Lower Bounds from Algebraic Circuit Complexity. Theory Comput. 17, 1–88.

Nicola Galesi & Massimo Lauria (2010). Optimality of size-degree tradeoffs for polynomial calculus. ACM Trans. Comput. Log. 12(1), 4:1–4:22.

D. Grigoriev (2001). Complexity of Positivstellensatz proofs for the knapsack. Computational Complexity 10(2), 139–154.

Dima Grigoriev (1998). Tseitin’s Tautologies and Lower Bounds for Nullstellensatz Proofs. In

*Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS’98)*, 648–652.Dima Grigoriev & Edward A. Hirsch (2003). Algebraic proof systems over formulas. Theoretical Computer Science 303(1), 83 – 102.

Joshua A. Grochow & Toniann Pitassi (2018). Circuit Complexity, Proof Complexity, and Polynomial Identity Testing: The Ideal Proof System. J. ACM 65(6), 37:1–37:59.

R. Impagliazzo, P. Pudlák & J. Sgall (1999). Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Computational Complexity 8(2), 127–144.

Russell Impagliazzo, Sasank Mouli & Toniann Pitassi (2020). The Surprising Power of Constant Depth Algebraic Proofs. In

*Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS’20)*, 591–603.Russell Impagliazzo, Sasank Mouli & Toniann Pitassi (2022). Lower bounds for Polynomial Calculus with extension variables over finite fields. Electron. Colloquium Comput. Complex. TR22-038. URL https://www.iso.ort/tandard/56426.html

Daniela Kaufmann, Paul Beame, Armin Biere & Jakob Nordstr öm (2022). 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).

Daniela Kaufmann & Armin Biere (2020). Nullstellensatz-Proofs for Multiplier Verification. In

*Proceedings of the 22nd International Workshop on Computer Algebra in Scientific Computing (CASC’20)*, 368–389.Daniela Kaufmann, Armin Biere & Manuel Kauers (2019). Verifying Large Multipliers by Combining SAT and Computer Algebra. In Proceedings of the 2019 Formal Methods in Computer Aided Design (FMCAD’19), 28–36.

Daniela Kaufmann, Armin Biere & Manuel Kauers (2020). From DRUP to PAC and Back. In Proceedings of the 2020 Design, Automation & Test in Europe Conference & Exhibition (DATE’20), 654–657

T.Y Lam & K.H Leung (2000). On Vanishing Sums of Roots of Unity. Journal of Algebra 224(1), 91–109.

J. Lasserre (2001). An explicit exact SDP relaxation for nonlinear 0-1 programs.

*Integer Programming and Combinatorial Optimization*293–303.Massimo Lauria & Jakob Nordström (2017). Graph Colouring is Hard for Algorithms Based on Hilbert’s Nullstellensatz and Gröbner Bases. In

*Proceedings of the 32nd Computational Complexity Conference*(CCC’17), volume 79, 2:1–2:20.Troy Lee, Anupam Prakash, Ronald de Wolf & Henry Yuen (2016). On the sum-of-squares degree of symmetric quadratic functions. In Proceedings of the 31st Conference on Computational Complexity (CCC’16), volume 50 of LIPIcs. Leibniz Int. Proc. Inform., Art. No. 17, 31.

João. Marques-Silva & Karem A. Sakallah (1999). GRASP: A search algorithm for propositional satisfiability. Computers, IEEE Transactions on 48(5), 506–521.

M.W. Moskewicz, C.F. Madigan, Y. Zhao, L. Zhang & S. Malik (2001). Chaff: Engineering an efficient SAT solver. In Proceedings of the 38th annual Design Automation Conference (DAC’01), 530–535.

Pablo A. Parrilo (2003). Semidefinite programming relaxations for semialgebraic problems. Mathematical programming 96(2), 293–320.

Aaron Potechin (2020). Sum of Squares Bounds for the Ordering Principle. In

*35th Computational Complexity Conference*(CCC’20), volume 169 of LIPIcs, 38:1–38:37.Susanna F. de Rezende, Massimo Lauria, Jakob Nordström & Dmitry Sokolov (2021). The Power of Negative Reasoning. In

*Proceedings of the 36th Computational Complexity Conference (CCC’21)*, volume 200 of LIPIcs, 40:1–40:24.Grant Schoenebeck (2008). Linear Level Lasserre Lower Bounds for Certain k-CSPs. In

*49th Annual IEEE Symposium on Foundations of Computer Science (FOCS ’08)*, 593–602.Roman Smolensky (1987). Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity. In

*Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC’87)*, 77–82. ACM.Dmitry Sokolov (2020). (Semi)Algebraic proofs over {±1} variables. In

*Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC’20)*.Madhur Tulsiani (2009). CSP gaps and reductions in the Lasserre hierarchy. In

*Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC’09)*, 303–312. ACM.

## Acknowledgements

The authors would like to thank Albert Atserias for fruitful discussions. The first author was supported by the Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033, Spain [grant numbers PID2019-109137GB-C21, PID2019-109137GB-C22, and IJC2018-035334-I].

## Funding

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Bonacina, I., Galesi, N. & Lauria, M. On vanishing sums of roots of unity in polynomial calculus and sum-of-squares.
*comput. complex.* **32**, 12 (2023). https://doi.org/10.1007/s00037-023-00242-z

Received:

Published:

DOI: https://doi.org/10.1007/s00037-023-00242-z