×

Polynomial root separation. (English) Zbl 1205.11032

The authors discuss recent advances and prove new results in the problem of estimating the minimal distance between two distinct roots of a polynomial with integral coefficients. In particular, they consider the above problem for monic cubic polynomials which is related to the Hall conjecture on small values of \(x^3-y^2\) and to the \(abc\)-conjecture.

MSC:

11C08 Polynomials in number theory
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birch B. J., Norske Vid. Selsk. Forh. (Trondheim) 38 pp 65–
[2] Bombieri E., Michigan Math. J. 33 pp 83–
[3] Bugeaud Y., Publ. Math. Debrecen 65 pp 305–
[4] DOI: 10.1017/CBO9780511542886 · doi:10.1017/CBO9780511542886
[5] DOI: 10.1017/S0013091503000257 · Zbl 1071.11016 · doi:10.1017/S0013091503000257
[6] Davenport H., Norske Vid. Selsk. Forh. (Trondheim) 38 pp 86–
[7] DOI: 10.1007/10722028_2 · doi:10.1007/10722028_2
[8] Evertse J.-H., Publ. Math. Debrecen 65 pp 323–
[9] DOI: 10.1016/j.jnt.2003.08.006 · Zbl 1055.11046 · doi:10.1016/j.jnt.2003.08.006
[10] Mahler K., Michigan Math. J. 11 pp 257–
[11] DOI: 10.1007/978-3-7091-3406-1_16 · doi:10.1007/978-3-7091-3406-1_16
[12] Mignotte M., RAIRO Anal. Numér. 13 pp 181–
[13] DOI: 10.5802/aif.1907 · Zbl 1014.12002 · doi:10.5802/aif.1907
[14] Schmidt W. M., Lecture Notes in Math. 1467, in: Diophantine Approximations and Diophantine Equations (1991) · Zbl 0754.11020 · doi:10.1007/BFb0098246
[15] DOI: 10.1016/j.jsc.2006.06.003 · Zbl 1158.12300 · doi:10.1016/j.jsc.2006.06.003
[16] Sprindžuk V. G., Lecture Notes in Math. 1559, in: Classical Diophantine Equations (1993) · doi:10.1007/BFb0073786
[17] Uchiyama S., Tsukuba J. Math. 6 pp 215–
[18] Wirsing E., J. Reine Angew Math. 206 pp 67–
[19] Zannier U., Acta Arith. 71 pp 107–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.