## 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.

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## 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].

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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

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