Quantitative height bounds under splitting conditions

Fili, Paul; Pottmeyer, Lukas

doi:10.1090/tran/7656

Quantitative height bounds under splitting conditions
HTML articles powered by AMS MathViewer

by Paul A. Fili and Lukas Pottmeyer PDF

Trans. Amer. Math. Soc. 372 (2019), 4605-4626 Request permission

Abstract:

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions such as being totally real or $p$-adic, improving on earlier work of Bombieri and Zannier in the totally $p$-adic case. These bounds applied as the degree of the algebraic number over the rationals tended towards infinity. In this paper, we use discrete energy approximation techniques on the Berkovich projective line to make the dependence on the degree in these bounds explicit, and we establish lower bounds for algebraic numbers which depend only on local properties of the numbers.

References

Matthew Baker, A lower bound for average values of dynamical Green’s functions, Math. Res. Lett. 13 (2006), no. 2-3, 245–257. MR 2231115, DOI 10.4310/MRL.2006.v13.n2.a6
Matthew Baker, An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves, $p$-adic geometry, Univ. Lecture Ser., vol. 45, Amer. Math. Soc., Providence, RI, 2008, pp. 123–174. MR 2482347, DOI 10.1090/ulect/045/04
Matthew Baker and Robert Rumely, Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, vol. 159, American Mathematical Society, Providence, RI, 2010. MR 2599526, DOI 10.1090/surv/159
Enrico Bombieri and Umberto Zannier, A note on heights in certain infinite extensions of $\Bbb Q$, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001), 5–14 (2002) (English, with English and Italian summaries). MR 1898444
E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), no. 4, 391–401. MR 543210, DOI 10.4064/aa-34-4-391-401
Artūras Dubickas and Michael J. Mossinghoff, Auxiliary polynomials for some problems regarding Mahler’s measure, Acta Arith. 119 (2005), no. 1, 65–79. MR 2163518, DOI 10.4064/aa119-1-5
Charles Favre and Juan Rivera-Letelier, Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), no. 2, 311–361 (French, with English and French summaries). MR 2221116, DOI 10.1007/s00208-006-0751-x
Charles Favre and Juan Rivera-Letelier, Corrigendum to: “Quantitative uniform distribution of points of small height on the projective line” (French) [Math. Ann. 335 (2006), no. 2, 311–361; MR2221116], Math. Ann. 339 (2007), no. 4, 799–801 (French). MR 2341901, DOI 10.1007/s00208-007-0130-2
Paul Fili and Clayton Petsche, Energy integrals over local fields and global height bounds, Int. Math. Res. Not. IMRN 5 (2015), 1278–1294. MR 3340356, DOI 10.1093/imrn/rnt250
Valérie Flammang, Georges Rhin, and Jean-Marc Sac-Épée, Integer transfinite diameter and polynomials with small Mahler measure, Math. Comp. 75 (2006), no. 255, 1527–1540. MR 2219043, DOI 10.1090/S0025-5718-06-01791-1
K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262. MR 166188
C. Petsche, The height of algebraic units in local fields, 2003. Unpublished note, available at http://people.oregonstate.edu/~petschec/HAULF.pdf.
Lukas Pottmeyer, Heights of points with bounded ramification, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 3, 965–981. MR 3445206
Robert S. Rumely, Capacity theory on algebraic curves, Lecture Notes in Mathematics, vol. 1378, Springer-Verlag, Berlin, 1989. MR 1009368, DOI 10.1007/BFb0084525
A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith. 24 (1973), 385–399. MR 360515, DOI 10.4064/aa-24-4-385-399
Paul Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), no. 1, 81–95. MR 1367580, DOI 10.4064/aa-74-1-81-95