##
**Counting solutions to trinomial Thue equations: a different approach.**
*(English)*
Zbl 0995.11025

Let \(F(x,y)\) be an irreducible form in \(\mathbb{Z}[x,y]\) with degree \(n\geq 3\), having exactly three non-zero coefficients. The author studies the number \(N_F\) of “regular solutions” to the trinomial Thue equation \(F(x,y)= \pm 1\) (a solution \((x,y)\) is “regular” if \(x\neq 0\), \(y>0\) and \(y\neq|x|\)). By a result of E. Bombieri and M. W. Schmidt on Thue (not necessarily trinomial) equations [Invent. Math. 88, 69-81 (1987; Zbl 0614.10018)], this number is at most constant times \(n\) and the constant can be taken to be 430 if \(n\) is sufficiently large. From the specialized to trinomial Thue equations work of J. Mueller and M. W. Schmidt [J. Reine Angew. Math. 379, 76-99 (1987; Zbl 0613.10019)], it follows that \(N_F\) is bounded by an effective but not explicit universal constant (independent from \(F\) and \(n\)). An explicit general result on Thue-Mahler equations over algebraic number fields, due to J.-H. Evertse [J. Reine Angew. Math. 482, 121-149 (1997; Zbl 0861.11021)], implies the explicit, but very large and dependent on \(n\), upper bound \(N_F\leq 5\cdot 10^6 n\).

The main result of the paper under review implies that, if \(n\geq 5\), then \(N_F\leq 81\) and, on specializing the range and parity of \(n\), the author obtains far better bounds; for example, if \(17\leq n\leq 37\) and \(n\) is odd, then \(N_F\leq 18\). The various specializations for \(n\) are explicitly stated in the paper.

As in the above mentioned papers of Bombieri, Mueller and Schmidt, the so-called “gap principle” and “Thue-Siegel principle” are basic tools. The approach of the present paper mainly differs from that of Bombieri-Mueller-Schmidt in the initial step: To each nontrivial solution \((x,y)= (p,q)\) these authors associate the closest to \(p/q\) root \(\omega\) (real or complex) of \(f(x):= F(x,1)\) and examine how well \(p/q\) approximates \(\omega\), while this author makes the following (we quote from his introduction): “[…] we associate to \((p,q)\) either a real root of \(f(x)\) or a real root of \(f'(x)\) […]. We call the set of all these roots exceptional set of \(F\). If \((p,q)\) is associated with an exceptional point \(\tau\), we again have the question: how well does \(p/q\) approximate the (real) number \(\tau\)? The method used here to solve the approximation problem is quite different […]. We work with polynomials with real coefficients, and we regard a given Thue trinomial as a member of a 1-parameter family of real trinomials. In this family we then select one or two trinomials called maximal. These trinomials have two key properties: first, the approximation problem is readily solved for these trinomials (and the solution is “almost” sharp); second, the solution for these trinomials is “maximal” for all trinomials in the family. In this way we solve the approximation problem for an arbitrary Thue trinomial.”

The author claims that his method can be extended to Thue tetranomials (i.e. those \(F\) with four nonzero coefficients) in a similar way.

The main result of the paper under review implies that, if \(n\geq 5\), then \(N_F\leq 81\) and, on specializing the range and parity of \(n\), the author obtains far better bounds; for example, if \(17\leq n\leq 37\) and \(n\) is odd, then \(N_F\leq 18\). The various specializations for \(n\) are explicitly stated in the paper.

As in the above mentioned papers of Bombieri, Mueller and Schmidt, the so-called “gap principle” and “Thue-Siegel principle” are basic tools. The approach of the present paper mainly differs from that of Bombieri-Mueller-Schmidt in the initial step: To each nontrivial solution \((x,y)= (p,q)\) these authors associate the closest to \(p/q\) root \(\omega\) (real or complex) of \(f(x):= F(x,1)\) and examine how well \(p/q\) approximates \(\omega\), while this author makes the following (we quote from his introduction): “[…] we associate to \((p,q)\) either a real root of \(f(x)\) or a real root of \(f'(x)\) […]. We call the set of all these roots exceptional set of \(F\). If \((p,q)\) is associated with an exceptional point \(\tau\), we again have the question: how well does \(p/q\) approximate the (real) number \(\tau\)? The method used here to solve the approximation problem is quite different […]. We work with polynomials with real coefficients, and we regard a given Thue trinomial as a member of a 1-parameter family of real trinomials. In this family we then select one or two trinomials called maximal. These trinomials have two key properties: first, the approximation problem is readily solved for these trinomials (and the solution is “almost” sharp); second, the solution for these trinomials is “maximal” for all trinomials in the family. In this way we solve the approximation problem for an arbitrary Thue trinomial.”

The author claims that his method can be extended to Thue tetranomials (i.e. those \(F\) with four nonzero coefficients) in a similar way.

Reviewer: Nikos Tzanakis (Iraklion)

### MSC:

11D59 | Thue-Mahler equations |

11D45 | Counting solutions of Diophantine equations |

11J68 | Approximation to algebraic numbers |

### Keywords:

trinomial Thue equation; number of solutions of Thue equations; Thue-Siegel principle; gap principle### References:

[1] | A. Baker, Rational approximations to certain algebraic numbers, Proc. London Math. Soc. (3) 14 (1964), 385 – 398. · Zbl 0131.29102 · doi:10.1112/plms/s3-14.3.385 |

[2] | A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms, Philos. Trans. Roy. Soc. London Ser. A 263 (1967/1968), 173 – 191. , https://doi.org/10.1098/rsta.1968.0010 A. Baker, Contributions to the theory of Diophantine equations. II. The Diophantine equation \?²=\?³+\?, Philos. Trans. Roy. Soc. London Ser. A 263 (1967/1968), 193 – 208. · Zbl 0157.09801 · doi:10.1098/rsta.1968.0011 |

[3] | Enrico Bombieri, On the Thue-Siegel-Dyson theorem, Acta Math. 148 (1982), 255 – 296. · Zbl 0505.10015 · doi:10.1007/BF02392731 |

[4] | Enrico Bombieri and Julia Mueller, On effective measures of irrationality for \root\?\of{\?/\?} and related numbers, J. Reine Angew. Math. 342 (1983), 173 – 196. · Zbl 0516.10024 |

[5] | E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. 88 (1987), no. 1, 69 – 81. · Zbl 0614.10018 · doi:10.1007/BF01405092 |

[6] | Davenport, H. and Roth, K. F. (1955). Rational approximations to algebraic numbers. Mathematika 3, 160-167. · Zbl 0066.29302 |

[7] | Dyson, F. J. (1947). The approximation to algebraic numbers by rationals. Acta Math. 79, 225-240. · Zbl 0030.02101 |

[8] | J.-H. Evertse, Upper bounds for the numbers of solutions of Diophantine equations, Mathematical Centre Tracts, vol. 168, Mathematisch Centrum, Amsterdam, 1983. · Zbl 0517.10016 |

[9] | N. I. Fel\(^{\prime}\)dman, An improvement of the estimate of a linear form in the logarithms of algebraic numbers, Mat. Sb. (N.S.) 77 (119) (1968), 423 – 436 (Russian). |

[10] | A. O. Gel\(^{\prime}\)fond, Transcendental and algebraic numbers, Translated from the first Russian edition by Leo F. Boron, Dover Publications, Inc., New York, 1960. |

[11] | Joseph Louis Lagrange, Oeuvres. Tome 2, Georg Olms Verlag, Hildesheim-New York, 1973. Publiées par les soins de J.-A. Serret; Nachdruck der Ausgabe Paris 1868. |

[12] | D. J. Lewis and K. Mahler, On the representation of integers by binary forms, Acta Arith. 6 (1960/1961), 333 – 363. · Zbl 0102.03601 |

[13] | Kurt Mahler, On Thue’s theorem, Math. Scand. 55 (1984), no. 2, 188 – 200. · Zbl 0544.10014 · doi:10.7146/math.scand.a-12075 |

[14] | Maurice Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992. Translated from the French by Catherine Mignotte. · Zbl 0741.11002 |

[15] | Julia Mueller, Counting solutions of |\?\?^{\?}-\?\?^{\?}|\le \?, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 503 – 513. · Zbl 0632.10014 · doi:10.1093/qmath/38.4.503 |

[16] | Julia Mueller and W. M. Schmidt, Trinomial Thue equations and inequalities, J. Reine Angew. Math. 379 (1987), 76 – 99. · Zbl 0613.10019 · doi:10.1515/crll.1987.379.76 |

[17] | J. Mueller and W. M. Schmidt, Thue’s equation and a conjecture of Siegel, Acta Math. 160 (1988), no. 3-4, 207 – 247. · Zbl 0655.10016 · doi:10.1007/BF02392276 |

[18] | Roth, K. F. (1955). Rational approximations to algebraic numbers, Mathematika 2, 1-20. · Zbl 0064.28501 |

[19] | Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. · Zbl 0754.11020 |

[20] | Schneider, T. (1957). Einfuhrung in die transzendenten Zahlen, Springer-Verlag, Berlin. |

[21] | Siegel, C. L. (1929). Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss., Math. Phys. Kl., Nr. 1=Ges. Abh. I., 209-266. · JFM 56.0180.05 |

[22] | C. L. Stewart, On the number of solutions of polynomial congruences and Thue equations, J. Amer. Math. Soc. 4 (1991), no. 4, 793 – 835. · Zbl 0744.11016 |

[23] | Thue, A. (1909). Über Annäherungswerte algebraischer Zahlen. J. Reine Angew. Math. 135, 284-305. |

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.