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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.MFCS.2022.23

Author Details

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

Funding

The first author was supported by the MICIN grants PID2019-109137GB-C22 and IJC2018-035334-I, and partially by the grant PID2019-109137GB-C21.

Acknowledgements

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

References

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