Abstract
For a polynomial f, a weighted sum-of-squares representation (SOS) has the form \(f = \sum_{i\in [s]} c_i f_i^2\), where the weights \(c_i\) are field elements. The size of the representation is the number of monomials that appear across the \(f_i\)'s. Its minimum across all such decompositions is called the support-sum S(f) of f.
For a univariate polynomial f of degree d of full support, a lower bound for the support-sum is \(S(f) \ge \sqrt d\). We show that the existence of an explicit univariate polynomial f with support-sum just slightly larger than the lower bound, that is, \(S(f) \ge d^{0.5+\varepsilon}\), for some \(\varepsilon > 0\), implies that VP \(\ne\) VNP, the major open problem in algebraic complexity. In fact, our proof works for some subconstant functions \(\varepsilon(d) > 0\) as well. We also consider the sum-of-cubes representation (SOC) of polynomials. We show that an explicit hard polynomial implies both blackbox-PIT is in P, and VP \(\neq\) VNP.
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Acknowledgements
P.D. is supported by the project Foundation of Lattice-based Cryptography, funded by NUS-NCS Joint Laboratory for Cyber Security, Singapore. This work was mostly done when P.D. was a research scholar at CMI, and a visiting scholar at CSE, IIT Kanpur, funded by Google PhD Fellowship (2018-2022). N.S. thanks the funding support from DST (SJF/MSA-01/2013-14), DST-SERB (CRG/2020/000045), and N.Rama.Rao Chair. Thanks to Manindra Agrawal for many useful discussions to optimize the SOS representations; to J. Maurice Rojas for several comments; to Arkadev Chattopadhyay for organizing a TIFR Seminar on this work. T.T. thanks DFG for the funding (grants TH 472/5-1 and TH 472/5-2), and CSE, IIT Kanpur for the hospitality
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Dutta, P., Saxena, N. & Thierauf, T. Weighted Sum-of-Squares Lower Bounds for Univariate Polynomials Imply \(\text{VP} \neq \text{VNP}\). comput. complex. 33, 3 (2024). https://doi.org/10.1007/s00037-024-00249-0
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DOI: https://doi.org/10.1007/s00037-024-00249-0