The Diophantine equation f(x) = g(y)
نویسندگان
چکیده
have finitely or infinitely many solutions in rational integers x and y? Due to the classical theorem of Siegel (see Theorem 10.1 below), the finiteness problem for (1), and even for a more general equation F (x, y) = 0 with F (x, y) ∈ Z[x, y], is decidable (). One has to: • decompose the polynomial F (x, y) into Q-irreducible factors; • for those factors which are not Q-reducible, determine the genus g and the number d of points at infinity of the corresponding plane curve; • for the factors with g = 0 and d ≤ 2 determine whether the corresponding equation has finitely or infinitely many integral solutions (see [4, Section 1]).
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