Fujita’s Program and Rational Points
نویسندگان
چکیده
A classical theme in mathematics is the study of integral solutions of diophantine equations, that is, equations with integral coefficients. The main problems are – decide the existence (or nonexistence) of solutions; – find (some or all) solutions; – describe (qualitatively or quantitatively) the set of solutions. Even the first of these questions is difficult, in full generality. As we know from logic, it is, in a sense, equivalent to all (formal) mathematics: for a given formal language, there exists one (nonhomogeneous) polynomial f(t, x1, ..., xn) (with Z-coefficients) such that the Statement #t is provable in this language iff the equation f(t, x1, ..., xn) = 0 has a solution with (x1, ..., xn) ∈ Z. Thus, at least theoretically, one can convert a problem in any field of mathematics, e.g. topology, to a problem in number theory. This is convincing evidence that general diophantine equations are extraordinarily complex.
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