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**The algorithmic resolution of Diophantine equations.**
*(English)*
Zbl 0907.11001

London Mathematical Society Student Texts. 41. Cambridge: Cambridge University Press. xvi, 243 p. (1998).

The constructive theory of Diophantine equations has undergone a very fast development in the last two decades. About ten years ago B. de Weger’s book [Algorithms for Diophantine equations, Centre for Mathematics and Computer Science Amsterdam, CWI-Tract 65 (1989; Zbl 0687.10013)] collected the main techniques of the 1980’s. The present book gives a useful, up to date and very interesting summary of the latest developments.

Part 1 of the book starts with introducing classical techniques like local methods and Skolem’s method. The theory of ternary quadratic forms is described, as well as continued fractions and the LLL basis reduction algorithm, which play an important role in the subsequent parts of the book.

Part 2 deals with practical algorithms for the resolution of Thue equations according to the methods of N. Tzanakis and B. de Weger [J. Number Theory 31, 99–132 (1989; Zbl 0657.10014)] and Y. Bilu and G. Hanrot [J. Number Theory 60, 373–392 (1996; Zbl 0867.11017)]. The theory of integral points on elliptic curves is introduced following Mordell’s book. The algorithm of N. Tzanakis and B. de Weger [Compos. Math. 84, 223–288 (1992; Zbl 0773.11023)] for the resolution of Thue-Mahler equations is detailed.

The effective theory of \(S\)-unit equations is investigated along the lines first shown by K. Győry [cf. e.g. Comment. Math. Helv. 54, 583–600 (1979; Zbl 0437.12004)] as well as practical methods for the resolution of them using reduction and sieve methods. (The book mentions but unfortunately does not include, K. Wildanger’s enumeration algorithm [Ph. D. Thesis, Technische Univ. Berlin (1997)].)

The theory of unit equations is then applied to triangularly connected decomposable form equations as introduced by K. Győry [Ann. Acad. Sci. Fenn., Ser. A I 4, 341–355 (1979; Zbl 0419.10021)]. The algorithm of N. Smart [Math. Comput. 64, 819–840 (1995; Zbl 0831.11027)] that enables one to solve explicitly such equations of low degree is also detailed.

These ideas are then applied to discriminant form equations. The method of N. Smart [Publ. Math. 43, 29–39 (1993; Zbl 0796.11047)] is described for the resolution of a discriminant form equation of Mahler-type in a quartic field.

The algorithm of I. Gaál, A. Pethő and M. Pohst [J. Symb. Comput. 16, 563–584 (1993; Zbl 0808.11023); J. Number Theory 57, 90–104 (1996; Zbl 0853.11023)] is then detailed, which gives a much more efficient method for the resolution of discriminant form equations (or equivalently index form equations) in any quartic field. Some algorithms for the resolution of index form equations in certain fields of higher degree are also mentioned.

Part 3 includes the theory of rational points on elliptic curves [cf. e.g. J. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press (1992; Zbl 0758.14042)]. The application of the Mordell-Weil theorem is shown with hints to the techniques developed by S. Siksek [Rocky Mt. J. Math. 25, 1501–1538 (1995; Zbl 0852.11028)].

One of the central topics of this part is the theory of integral points on elliptic curves. The book includes a survey of the important tools and gives many references to recent achievements like S. David [Mém. Soc. Math. Fr. 62 (1995; Zbl 0859.11048)] who gave the first lower bounds for linear forms in elliptic logarithms, R. Stroeker and N. Tzanakis [Acta Arith. 67, 177–196 (1994; Zbl 0805.11026)] and J. Gebel, A. Pethő and H. Zimmer [Acta Arith. 68, 171–192 (1994; Zbl 0816.11019)] who independently worked out efficient algorithms for solving elliptic equations in integers, and N. Smart [Math. Proc. Camb. Philos. Soc. 116, 391–399 (1994; Zbl 0817.11031)] who considered \(S\)-integral points on elliptic curves.

Finally, curves of genus larger than one are investigated, with emphasis on hyperelliptic and superelliptic equations, as well as Fermat curves and Catalan’s equation.

The book gives many interesting historical notes throughout and includes exercises that may introduce the reader to modern techniques. The book will surely become useful both for interested students and researchers of the area.

Part 1 of the book starts with introducing classical techniques like local methods and Skolem’s method. The theory of ternary quadratic forms is described, as well as continued fractions and the LLL basis reduction algorithm, which play an important role in the subsequent parts of the book.

Part 2 deals with practical algorithms for the resolution of Thue equations according to the methods of N. Tzanakis and B. de Weger [J. Number Theory 31, 99–132 (1989; Zbl 0657.10014)] and Y. Bilu and G. Hanrot [J. Number Theory 60, 373–392 (1996; Zbl 0867.11017)]. The theory of integral points on elliptic curves is introduced following Mordell’s book. The algorithm of N. Tzanakis and B. de Weger [Compos. Math. 84, 223–288 (1992; Zbl 0773.11023)] for the resolution of Thue-Mahler equations is detailed.

The effective theory of \(S\)-unit equations is investigated along the lines first shown by K. Győry [cf. e.g. Comment. Math. Helv. 54, 583–600 (1979; Zbl 0437.12004)] as well as practical methods for the resolution of them using reduction and sieve methods. (The book mentions but unfortunately does not include, K. Wildanger’s enumeration algorithm [Ph. D. Thesis, Technische Univ. Berlin (1997)].)

The theory of unit equations is then applied to triangularly connected decomposable form equations as introduced by K. Győry [Ann. Acad. Sci. Fenn., Ser. A I 4, 341–355 (1979; Zbl 0419.10021)]. The algorithm of N. Smart [Math. Comput. 64, 819–840 (1995; Zbl 0831.11027)] that enables one to solve explicitly such equations of low degree is also detailed.

These ideas are then applied to discriminant form equations. The method of N. Smart [Publ. Math. 43, 29–39 (1993; Zbl 0796.11047)] is described for the resolution of a discriminant form equation of Mahler-type in a quartic field.

The algorithm of I. Gaál, A. Pethő and M. Pohst [J. Symb. Comput. 16, 563–584 (1993; Zbl 0808.11023); J. Number Theory 57, 90–104 (1996; Zbl 0853.11023)] is then detailed, which gives a much more efficient method for the resolution of discriminant form equations (or equivalently index form equations) in any quartic field. Some algorithms for the resolution of index form equations in certain fields of higher degree are also mentioned.

Part 3 includes the theory of rational points on elliptic curves [cf. e.g. J. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press (1992; Zbl 0758.14042)]. The application of the Mordell-Weil theorem is shown with hints to the techniques developed by S. Siksek [Rocky Mt. J. Math. 25, 1501–1538 (1995; Zbl 0852.11028)].

One of the central topics of this part is the theory of integral points on elliptic curves. The book includes a survey of the important tools and gives many references to recent achievements like S. David [Mém. Soc. Math. Fr. 62 (1995; Zbl 0859.11048)] who gave the first lower bounds for linear forms in elliptic logarithms, R. Stroeker and N. Tzanakis [Acta Arith. 67, 177–196 (1994; Zbl 0805.11026)] and J. Gebel, A. Pethő and H. Zimmer [Acta Arith. 68, 171–192 (1994; Zbl 0816.11019)] who independently worked out efficient algorithms for solving elliptic equations in integers, and N. Smart [Math. Proc. Camb. Philos. Soc. 116, 391–399 (1994; Zbl 0817.11031)] who considered \(S\)-integral points on elliptic curves.

Finally, curves of genus larger than one are investigated, with emphasis on hyperelliptic and superelliptic equations, as well as Fermat curves and Catalan’s equation.

The book gives many interesting historical notes throughout and includes exercises that may introduce the reader to modern techniques. The book will surely become useful both for interested students and researchers of the area.

Reviewer: István Gaál (Debrecen)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11Y50 | Computer solution of Diophantine equations |

11D25 | Cubic and quartic Diophantine equations |

11D57 | Multiplicative and norm form equations |

11D75 | Diophantine inequalities |

11G05 | Elliptic curves over global fields |

14G05 | Rational points |

11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |

11J86 | Linear forms in logarithms; Baker’s method |

### Keywords:

Thue equations; \(S\)-unit equations; decomposable form equations; discriminant form equations; elliptic curves; Mordell-Weil theorem; curves of genus greater than one; resolution of Thue-Mahler equations; sieve methods; index form equations; rational points; hyperelliptic curves; superelliptic equations; Fermat curves; Catalan’s equation### Citations:

Zbl 0687.10013; Zbl 0657.10014; Zbl 0867.11017; Zbl 0773.11023; Zbl 0437.12004; Zbl 0419.10021; Zbl 0831.11027; Zbl 0796.11047; Zbl 0808.11023; Zbl 0853.11023; Zbl 0758.14042; Zbl 0852.11028; Zbl 0859.11048; Zbl 0805.11026; Zbl 0816.11019; Zbl 0817.11031
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\textit{N. P. Smart}, The algorithmic resolution of Diophantine equations. Cambridge: Cambridge University Press (1998; Zbl 0907.11001)