Abstract
If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional restrictions, e.g., to find all roots that belong to a given finite dimensional k-subspace of K, more efficiently. For bounded characteristic, it runs in polynomial time, relative to factorizations over the constant field k and also provides a deterministic polynomial time irreducibility test. We also discuss applications to places of reducible reduction, when k is a global field, and to list decoding of Reed-Solomon codes.
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Acknowledgements
This research was funded by the Ministry for Business, Innvovation and Employment in New Zealand. I would also like to thank M. Esgin, V. Kuchta, S. Ruj, A. Sakzad and R. Steinfeld of the Trans-Tasman ZK group for questions that motivated part of this research and a number of referees for their very thorough and useful reports.
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Voloch, J.F. Factoring polynomials over function fields. Res. number theory 11, 5 (2025). https://doi.org/10.1007/s40993-024-00581-y
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DOI: https://doi.org/10.1007/s40993-024-00581-y