Simple families of Thue inequalities. (English) Zbl 0920.11041

Lettl, Günter; Pethő, Attila; Voutier, Paul

Trans. Am. Math. Soc. 351, No. 5, 1871-1894 (1999).

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

Keywords:

diophantine inequalities; Thue equations; method of hypergeometric functions; simple forms

References:

[1]	A. Baker, Rational approximations to \root3\of2 and other algebraic numbers, Quart. J. Math. Oxford Ser. (2) 15 (1964), 375 – 383. · Zbl 0222.10036
[2]	Michael A. Bennett, Effective measures of irrationality for certain algebraic numbers, J. Austral. Math. Soc. Ser. A 62 (1997), no. 3, 329 – 344. · Zbl 0880.11055
[3]	Jian Hua Chen, A new solution of the Diophantine equation \?²+1=2\?\(^{4}\), J. Number Theory 48 (1994), no. 1, 62 – 74. · Zbl 0814.11021
[4]	Jian Hua Chen and Paul Voutier, Complete solution of the Diophantine equation \?²+1=\?\?\(^{4}\) and a related family of quartic Thue equations, J. Number Theory 62 (1997), no. 1, 71 – 99. · Zbl 0869.11025
[5]	G. V. Chudnovsky, On the method of Thue-Siegel, Ann. of Math. (2) 117 (1983), no. 2, 325 – 382. · Zbl 0518.10038
[6]	J. H. E. Cohn, Equations with equivalent roots, Acta Arith. 34 (1977/78), no. 1, 37 – 41. · Zbl 0369.12013
[7]	David Easton, Effective irrationality measures for certain algebraic numbers, Math. Comp. 46 (1986), no. 174, 613 – 622. · Zbl 0586.10019
[8]	Marie-Nicole Gras, Familles d’unités dans les extensions cycliques réelles de degré 6 de \?, Théorie des nombres, Années 1984/85 – 1985/86, Fasc. 2, Publ. Math. Fac. Sci. Besançon, Univ. Franche-Comté, Besançon, 1986, pp. Exp. No. 2, 27 (French).
[9]	Andrew J. Lazarus, On the class number and unit index of simplest quartic fields, Nagoya Math. J. 121 (1991), 1 – 13. · Zbl 0719.11073
[10]	G. Lettl and A. Pethő, Complete solution of a family of quartic Thue equations, Abh. Math. Sem. Univ. Hamburg 65 (1995), 365 – 383. · Zbl 0853.11021
[11]	G. Lettl, A. Pethő and P. Voutier, On the arithmetic of simplest sextic fields and related Thue equations, Number Theory: Diophantine, Computational and Algebraic Aspects , Walter de Gruyter Publ. Co., 1998, 331-348. · Zbl 0923.11053
[12]	Falko Lorenz, Lineare Algebra. II, 2nd ed., Bibliographisches Institut, Mannheim, 1989 (German). · Zbl 0664.15001
[13]	Kevin S. McCurley, Explicit estimates for \?(\?;3,\?) and \?(\?;3,\?), Math. Comp. 42 (1984), no. 165, 287 – 296. · Zbl 0535.10044
[14]	Maurice Mignotte, Verification of a conjecture of E. Thomas, J. Number Theory 44 (1993), no. 2, 172 – 177. · Zbl 0780.11013
[15]	M. Mignotte, A. Pethő, and F. Lemmermeyer, On the family of Thue equations \?³-(\?-1)\?²\?-(\?+2)\?\?²-\?³=\?, Acta Arith. 76 (1996), no. 3, 245 – 269. · Zbl 0862.11028
[16]	Maurice Mignotte, Attila Pethő, and Ralf Roth, Complete solutions of a family of quartic Thue and index form equations, Math. Comp. 65 (1996), no. 213, 341 – 354. · Zbl 0853.11022
[17]	Attila Pethő, On the resolution of Thue inequalities, J. Symbolic Comput. 4 (1987), no. 1, 103 – 109. · Zbl 0625.10011
[18]	Attila Pethő, Complete solutions to families of quartic Thue equations, Math. Comp. 57 (1991), no. 196, 777 – 798. · Zbl 0738.11028
[19]	Olivier Ramaré and Robert Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), no. 213, 397 – 425. · Zbl 0856.11042
[20]	J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64 – 94. · Zbl 0122.05001
[21]	Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535 – 541. , https://doi.org/10.1090/S0025-5718-1988-0929551-0 René Schoof and Lawrence C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), no. 182, 543 – 556. · Zbl 0649.12007
[22]	Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137 – 1152. · Zbl 0307.12005
[23]	Emery Thomas, Complete solutions to a family of cubic Diophantine equations, J. Number Theory 34 (1990), no. 2, 235 – 250. · Zbl 0697.10011
[24]	P. M. Voutier, Rational approximations to \(\sqrt[3 ]{2}\) and other algebraic numbers revisited, Indag. Math. (to appear). · Zbl 1120.11027