Let F(X, Y) = Σmi=0Σnj=0ai,jXiYj be an absolutely irreducible polynomial in Z[X,K]. Suppose that m > 1,n > 2 and that the polynomial Σnj=0am,jYj reducible in Q[K], has n simple roots and an unique real root. Let L be a totally real number field and let (ξ,ζ)∈Ol×L be such that F(ξ,ζ) = 0. We give an upper bound for the absolute height H(ξ) which depends only upon the polynomial F. Our result may be viewed as a natural extension of Runge's method to arbitrary totally real number fields.
展开▼
