Abstract
Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly \( (s^{ {\rm \log} s}) \).
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Acknowledgements
We acknowledge the support from DFG Grant TH 472/5-1. We thank Nitin Saxena, Pranjal Dutta, Arpita Korwar, Sumanta Ghosh, Zeyu Guo, and Mrinal Kumar for helpful discussions. We thank Vishwas Bhargav for pointing out to us that the subresultant technique of computing GCD is suitable for the ABP-model. We thank Nutan Limaye for asking about the ABP-size for approximate power series roots. A.S. would like to thank the Institute of Theoretical Computer Science at Ulm University for the hospitality.
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Sinhababu , A., Thierauf, T. Factorization of Polynomials Given by Arithmetic Branching Programs. comput. complex. 30, 15 (2021). https://doi.org/10.1007/s00037-021-00215-0
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DOI: https://doi.org/10.1007/s00037-021-00215-0
Keywords
- Arithmetic Branching Program
- Multivariate Polynomial Factorization
- Hensel Lifting
- Newton Iteration
- Hardness vs Randomness