Linear polynomials in numbers of bounded degree
The main result of this paper isNEWLINENEWLINETheorem. Let \(f(X_0,X_1,\ldots, X_n)\) be a nonzero homogeneous polynomial with rational coefficients, and of total degree at most \(d\). Suppose that \((\alpha_0, \alpha_1, \ldots, \alpha_n)\) has algebraic components and is not a zero of \(f\). Then given NEWLINE\[NEWLINE B > d {d+n \choose n}+d, NEWLINE\]NEWLINE there are only finitely many points \((\beta_1, \ldots, \beta_n)\) with \([\mathbb Q(\beta_0, \beta_1, \ldots, \beta_n) : \mathbb Q ]\leq d\) and NEWLINE\[NEWLINE |\alpha_0+ \alpha_1,\beta_1 +\cdots+ \alpha_n \beta_n| < H(\beta_1, \ldots, \beta_n)^{-B}, NEWLINE\]NEWLINE where \( H(\beta_1, \ldots, \beta_n)\) is the absolute Weil height of the projective point \((1 : \beta_1 : \ldots : \beta_n)\).NEWLINENEWLINEThis result is close to a dual of a result of Philippon and Schlickewei about simultaneous approximation by algebraic \(n\)--tuples of bounded degree.NEWLINENEWLINEThe main tool in the proof is the Subspace Theorem.
scientific article; zbMATH DE number 6090461