Document Zbl 1456.11034

Thue Diophantine equations. (English) Zbl 1456.11034

Chakraborty, Kalyan (ed.) et al., Class groups of number fields and related topics. Collected papers presented at the first international conference, ICCGNFRT, Harish-Chandra Research Institute, Allahabad, India, September 4–7, 2017. Singapore: Springer. 25-41 (2020).

This survey presents the essentials concerning Thue (mainly) and Thue-Mahler equations. In this reviewer’ s opinion, it is very much appropriate for graduate students of Mathematics looking for their way in Number Theory and for any mathematician who would like to know about what is about these equations and what kind of Mathematics are involved in their study.
Description of the paper’s contents. Section 1: Basic definitions. The special case of Thue equation in which positive definite binary forms are involved, and CM-fields. General Thue equations and their relation to Diophantine Approximation. Thue-Siegel-Roth Theorem and Thue’s Theorem (without proof of course). Description of Thue’s method by a concrete example.
Section 2. Solving Thue equations by Baker’s method (use of linear forms in logarithms of algebraic numbers); a list of book references for further study is given. Thue equation and Siegel’s Unit equation. Lower bounds for linear forms in logarithms and Siege’s Unit equation.
Section 3. Families of Thue equations and finiteness of solutions to such families. Short description of the author’s research project jointly with Claude Levesque, and a related conjecture.
Section 4. A guide for further references.
For the entire collection see [Zbl 1444.11004].

Reviewer: Nikos Tzanakis (Iraklion)

Cited in 2 Documents

MSC:

11D59	Thue-Mahler equations
11J68	Approximation to algebraic numbers
11J86	Linear forms in logarithms; Baker’s method

Keywords:

Thue-Mahler equation; binary form; CM-field; Diophantine approximation; Siegel’s unit equation; linear form in logarithms; Baker’s method; Schmidt’s subspace theorem; family of Thue equations

Software:

OEIS

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