Algebraic integral representations by arbitrary forms

Extract

If(X₁, …, X_n), n ≥ 3, is a non-singular quadratic form with rational integral coefficients whose greatest common divisor is 1, then G. L. Watson [1] showed that f(x₁, …, x_n) = 1, for suitable algebraic integers x₁, …, x_n. In the present paper we extend this result to forms of arbitrary degree, with algebraic integers as coefficients (see Theorem 3). In fact we prove the stronger result (Theorem 2) that, if f(X₁, …, X_n) is any polynomial with relatively prime algebraic integers as coefficients, then f(x₁, …, x_n) is a unit, for suitable algebraic integers x₁, …, x_n. Unfortunately, our result is just an existence theorem. We cannot limit the size of the field which x₁, …, x_n generate, as Watson could.