On equal values of binary forms over finitely generated fields

B, Brindza; Á, Pintér [Pintér, Ákos (Számelmélet), szerző] Matematikai Intézet (DE / TTK)

Angol nyelvű Tudományos Szakcikk (Folyóiratcikk)
Megjelent: PUBLICATIONES MATHEMATICAE DEBRECEN 0033-3883 2064-2849 46 (3-4) pp. 339-347 1995
      As it was pointed out by Lang [4, p. 245] and others, certain finiteness results for diophantine equations over algebraic number fields can be extended, by using deep algebraic geometrical arguments, to rather general cases when the ground domain of unknowns is a finitely generated field or a finitely generated subring of it. The purpose of this paper is to establish a surprisingly elementary method, through a concrete equation, to obtain these kind of general results. Let f(X, Y) and g(X, Y) he binary forms (homogeneous polynomials in two variables) with complex coefficients of degree m and n, respectively. The binary form fg splits into linear factors (over C) and in the sequel, we suppose that the linear factors are pairwise non-proportional. Let K be a finitely generated subfield of C. Then K can be written in the form Q(z(1),.,z(q), u), where z(1),...,z(4) is a transcendence basis of K and we may assume without loss of generality that the element u is integral over the polynomial ring Z[z(1),...,z(q)].
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      2020-11-29 13:55