Abstract
The house of an algebraic integer is the maximum absolute value of its algebraic conjugates. Lower bounds for the house may be easier to prove than lower bounds for the height, as V. Dimitrov’s recent proof of the Schinzel–Zassenhaus conjecture suggests. We prove an analogue for the house of a recent conjecture by G. Rémond on lower bounds for the height in some radical extensions.
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Notes
Here and in what follows, we denote by \(\zeta _n\) a primitive nth root of unity.
This orthogonality relation is also one of the ingredient of an other result on height, see [4], proof of Theorem 1.
That is: \(\gamma _1^{a_1},\ldots ,\gamma _r^{a_r}\in (K^*)^N\) whenever \(\gamma _1^{a_1}\cdots \gamma _r^{a_r}\in (K^*)^N\) for some integers \(a_i\).
note that the extension \({\mathbb Q}^{\textrm{tr}}(i)/{\mathbb Q}^{\textrm{tr}}\) is essentially the only known example of this phenomenon, see [7].
Thus \(\alpha >1\) is an algebraic integer, \(\alpha ^{-1}\) is a conjugate of \(\alpha \), and the other conjugates lie on the unit circle.
References
Amoroso, F.: On a conjecture of G. Rémond. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15, 599–608 (2016)
Amoroso, F., David, S., Zannier, U.: On fields with the property (B). Proc. Am. Math. Soc. 142(6), 893–1910 (2014)
Amoroso, F., Dvornicich, R.: A lower bound for the height in abelian extensions. J. Number Theory 80(2), 260–272 (2000)
Bombieri, E., Zannier, U.: A note on heights in certain infinite extensions of \(\mathbb{Q} \). Rend. Mat. Acc. Lincei 9(12), 5–14 (2001)
Conrad, K.: The ring of integers in a radical extension. https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf
Dimitrov, V.: A proof of the Schinzel-Zassenhaus conjecture on polynomials. Preprint. https://arxiv.org/abs/1912.12545
Pottmeyer, L.: A note on extensions of \({{\mathbb{Q} }^{{\rm tr}}}\). J. Théor. Nombres Bordeaux 28, 735–742 (2016)
Rémond, G.: Généralisations du problème de Lehmer et applications à la conjecture de Zilber-Pink. Panor. Synthèses 52, Soc. Math. France, Paris (2017)
Schinzel, A.: On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith. 24, 385–399. Addendum; ibid. 26(1973), 329–361 (1973)
Acknowledgements
We are indebted to Gaël Rémond, who suggested to us a simple proof of Lemma 2.2 with a better (and optimal) constant, and to Lukas Pottmeyer for Remark 4.2.
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Amoroso, F. Lower bounds for the house in some radical extensions. Ramanujan J 63, 497–505 (2024). https://doi.org/10.1007/s11139-023-00771-9
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DOI: https://doi.org/10.1007/s11139-023-00771-9