Over the course of the last 15 years or so, Minhyong Kim has developed a method
\nfor making eff\u000bective use of the fundamental group to bound sets of solutions to hyperbolic
\nequations; his method opens a new avenue in the quest for an eff\u000bective version of the Mordell
\nconjecture. But although Kim’s approach has led to the construction of explicit bounds in
\nspecial cases, the problem of realizing the potential e\u000bffectivity of his methods remains a diff\u000ecult
\nand beautiful open problem. In the case of the unit equation, this problem may be approached
\nvia “motivic” methods. Using these methods we are able to describe an algorithm; its output
\nupon halting is provably the set of integral points, while its halting depends on conjectures.<\/p>\n","protected":false},"excerpt":{"rendered":"
Over the course of the last 15 years or so, Minhyong Kim has developed a method for making eff\u000bective use of the fundamental group to bound sets of solutions to hyperbolic equations; his method opens a new avenue in the…
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