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**Simple families of Thue inequalities.**
*(English)*
Zbl 0920.11041

Among the few general methods in diophantine approximation are Baker’s method of linear forms in logarithms and the method of hypergeometric functions introduced by Thue and Siegel (and also used very efficiently by Baker). It is well known that the second method has a range of application smaller than Baker’s, but is able to produce much better estimates than the application of the actual estimates on linear forms in logarithms. This paper is a very good illustration of this fact: the authors are able to apply the method of hypergeometric functions to three families of parametrized Thue inequalities, and their results are much better than those obtained earlier by Baker’s method. One should also notice that the method of hypergeometric functions does not need the determination of the arithmetic of the underlying number fields (which can be the most expensive part of the work when using Baker’s method, notice also that these three authors study this structure in the case of the sextic example in another paper – the other cases being already studied in earlier works). The families studied correspond to the three following forms:
\[
\begin{aligned} F &= X^3-tX^2Y-(t+3)XY^2-Y^3,\\ G &= X^4-tX^3Y-6X^2Y^2+tXY^3+Y^4 \quad\text{ and}\\ H &= X^6-2tX^5Y-5(t+3)X^4Y^2-20X^3Y^3+5tX^2Y^4+2(t+3)XY^5+Y^6. \end{aligned}
\]
Among other results, the authors solve completely the inequalities \(| H(x,y)| \leq 120t+323\) for \(t\geq 89\), and \(| G(x,y)| \leq 6t+7\) for \(t\geq 58\). (Notice that the conditions on \(t\) are inherent to this method.) They also show that these three families are the only examples of some kind of irreducible forms of degree \({}\geq 3\) which they call simple forms.

Reviewer: Maurice Mignotte (Strasbourg)

### MSC:

11J25 | Diophantine inequalities |

11D75 | Diophantine inequalities |

11D25 | Cubic and quartic Diophantine equations |

11D41 | Higher degree equations; Fermat’s equation |

11J82 | Measures of irrationality and of transcendence |

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\textit{G. Lettl} et al., Trans. Am. Math. Soc. 351, No. 5, 1871--1894 (1999; Zbl 0920.11041)

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### References:

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