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Hi,

I am interested in studying the Thue equation, where we are concerned with a binary form $F(x,y) = a_0 x^r + a_1 x^{r-1}y + \cdots + a_r y^r$ and solutions of the form $$F(x,y) = h$$ for some integer $h$. In particular, I am interested in works that investigate giving bounds on the number of solutions to the equation (necessarily finite by the Thue-Siegel-Roth Theorem), and on the size of the solutions. In particular I am interested in effective methods. Some papers in this field include:

C.L. Stewart, "On the number of solutions of polynomial congruences and Thue equations", Journal of the American Math. Soc., 4 (1991), 793-835.

E. Bombieri and W.M. Schmidt, "On Thue's Equation", Invent. Math, 88., (1987), 69-81.

I would like to request some additional papers on this subject, and if it exists, a good book on the subject matter.

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5 Answers 5

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Silverman's papers (Inv. Math vols. 66 and 74) bounding the number of solutions in terms of the Mordell-Weil rank of the Jacobians are very nice.

Baker's book on transcendence theory does the application to bounds for the heights of the solutions.

Thue's equation is mentioned in most books on diophantine geometry (Lang, Mordell, Bombieri-Gubler) but I don't think there is a book specifically devoted to it.

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    $\begingroup$ Thanks for the citation, Felipe. Those papers are effective in principle, but the constants are not worked out. One gets that the number of integer solutions is smaller than $C(F)^{1+r(F,h)}$, where $r(F,h)$ is the rank of the Mordell-Weil group of the Jacobian of the curve $F(x,y)=h$, but I did not make $C(F)$ explicit. Rob Gross and I worked out completely explicit (albeit huge) bounds for the number of $S$-integral points on elliptic curves, so it can be done! This is in $S$-Integer Points on Elliptic Curves, Pacific Jour. of Math 167 (1995), 263-288. $\endgroup$
    – Joe Silverman
    Commented Jun 5, 2011 at 0:07
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There is a fairly recent survey on (parametrized) Thue equations by C. Heuberger with many (recent) references:

Parametrized Thue Equations — A survey, Proceedings of the RIMS symposium “Analytic Number Theory and Surrounding Areas”, Kyoto, Oct 18–22, 2004, RIMS Kôkyûroku 1511, August 2006, 82–91

http://www.math.tugraz.at/~cheub/publications/thue-survey.pdf

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Chapter 7 of Nigel P Smart, The Algorithmic Resolution of Diophantine Equations, is entitled, Thue equations (and Chapter 8 is Thue-Mahler equations). This is in the part of the book devoted to methods using linear forms in logarithms.

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You may like this survey article of Prof. Michel Waldschmidt https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ProcHRI2017ThueEquations.pdf. It is relatively newer.

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In the case that the Mordell-Weil rank of the Jacobian of the curve is less than the genus, Lorenzini and Tucker used the Chabauty-Coleman method to give an upper bound on the number of rational points that depends only on the degree $r$.

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