Abstract
In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree \(n \ge 2\) and height bounded by \(H \ge 2\). The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.
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Acknowledgments
The authors would like to thank Igor E. Shparlinski for introducing them into this topic and for providing Proposition 1 which was their starting point, and also for his valuable comments on an early version of this paper. They also want to thank the referee for careful reading and useful comments.
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Communicated by J. Schoißengeier.
The research of A. Dubickas was supported by the Research Council of Lithuania Grant MIP-068/2013/LSS-110000-740. The research of M. Sha was supported by the Australian Research Council Grant DP130100237.
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Dubickas, A., Sha, M. Counting degenerate polynomials of fixed degree and bounded height. Monatsh Math 177, 517–537 (2015). https://doi.org/10.1007/s00605-014-0680-9
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DOI: https://doi.org/10.1007/s00605-014-0680-9