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**Diophantine equations over function fields.**
*(English)*
Zbl 0533.10012

London Mathematical Society Lecture Note Series, 96. Cambridge etc.: Cambridge University Press. x, 125 p. £7.95; $ 15.95 (1984).

Let \(K\) be an arbitrary finite extension of the rational function field \(k(z)\) over an algebraically closed field \(k\) of characteristic zero. In this book algorithms are provided for the effective determination of all the solutions in \(\mathcal O\), the ring of integral elements of \(K\) over \(k[z]\), of various general families of diophantine equations in two unknowns with coefficients in \(K\), such as

Thue equations \[ (X - \alpha_1Y)\cdots (X - \alpha_nY) = \mu, \] where \(n\geq 3\) and \(\alpha_1,\ldots,\alpha_n\) and \(\mu\) are elements of \(\mathcal O\) such that \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\) are distinct and \(\mu\neq 0\),

hyperelliptic equations \[ Y^2 = (X - \alpha_1)\cdots (X - \alpha_n) \] with \(n\geq 3\) and distinct elements \(\alpha_1,\ldots,\alpha_n\) of \(\mathcal O\), and

equations of the form \(F(X,Y)=0\) of genus zero and genus one types, where \(F\) denotes an absolutely irreducible polynomial with coefficients in \(K\).

Explicit bounds for the degrees of the solutions and criteria for the existence of infinitely many solutions of such equations are also given. Several worked out examples illustrate the author’s general method.

These results are obtained by appealing to a new fundamental inequality, first announced by the author in 1982; thus, the approach to the problem is completely different from those of previous authors [cf. C. F. Osgood, Mathematika 20, 4–15 (1973; Zbl 0269.10017), and W. M. Schmidt, J. Aust. Math. Soc., Ser. A 25, 385–422 (1978; Zbl 0389.10019)].

Furthermore, by extending the application of the fundamental inequality to fields of positive characteristic, the complete, effective resolution of the Thue and hyperelliptic equations in that circumstance is achieved. The author also applies the fundamental inequality in a different manner to find explicit bounds on the solutions in rational functions of certain general classes of superelliptic equations. It is remarked that, although the algorithms all apply to function fields in one variable, there is no difficulty in the extension to arbitrarily many variables, by means of a simple inductive argument.

This small but remarkable monograph presents a self-contained account of the author’s new and original approach to the subject.

Thue equations \[ (X - \alpha_1Y)\cdots (X - \alpha_nY) = \mu, \] where \(n\geq 3\) and \(\alpha_1,\ldots,\alpha_n\) and \(\mu\) are elements of \(\mathcal O\) such that \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\) are distinct and \(\mu\neq 0\),

hyperelliptic equations \[ Y^2 = (X - \alpha_1)\cdots (X - \alpha_n) \] with \(n\geq 3\) and distinct elements \(\alpha_1,\ldots,\alpha_n\) of \(\mathcal O\), and

equations of the form \(F(X,Y)=0\) of genus zero and genus one types, where \(F\) denotes an absolutely irreducible polynomial with coefficients in \(K\).

Explicit bounds for the degrees of the solutions and criteria for the existence of infinitely many solutions of such equations are also given. Several worked out examples illustrate the author’s general method.

These results are obtained by appealing to a new fundamental inequality, first announced by the author in 1982; thus, the approach to the problem is completely different from those of previous authors [cf. C. F. Osgood, Mathematika 20, 4–15 (1973; Zbl 0269.10017), and W. M. Schmidt, J. Aust. Math. Soc., Ser. A 25, 385–422 (1978; Zbl 0389.10019)].

Furthermore, by extending the application of the fundamental inequality to fields of positive characteristic, the complete, effective resolution of the Thue and hyperelliptic equations in that circumstance is achieved. The author also applies the fundamental inequality in a different manner to find explicit bounds on the solutions in rational functions of certain general classes of superelliptic equations. It is remarked that, although the algorithms all apply to function fields in one variable, there is no difficulty in the extension to arbitrarily many variables, by means of a simple inductive argument.

This small but remarkable monograph presents a self-contained account of the author’s new and original approach to the subject.

Reviewer: SaburĂ´ Uchiyama (Tsukuba)

### MSC:

11D88 | \(p\)-adic and power series fields |

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

11R58 | Arithmetic theory of algebraic function fields |

11D61 | Exponential Diophantine equations |

11D99 | Diophantine equations |

11J68 | Approximation to algebraic numbers |

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

14G05 | Rational points |