Abstract
We obtain upper bounds for the number of monic irreducible polynomials over \(\mathbb{Z}\) of a fixed degree n and a growing height H for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree n and height at most H. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant, improving previous results of I. E. Shparlinski (2010).
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Acknowledgement
The authors are grateful to Nicholas Katz for valuable discussions regarding several issues about discriminants of number fields and also for providing the second example of Section 6. The authors also would like to thank the referee for the careful reading of the paper and several valuable suggestions improving the exposition of the paper.
During the preparation of this work, A. O. was supported by the ARC Grant DP180100201 and I. S. was supported by the ARC Grant DP170100786.
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Dietmann, R., Ostafe, A. & Shparlinski, I.E. Discriminants of fields generated by polynomials of given height. Isr. J. Math. 260, 73–103 (2024). https://doi.org/10.1007/s11856-023-2557-x
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DOI: https://doi.org/10.1007/s11856-023-2557-x