involves constructing an auxiliary multivariate polynomial in an arbitrarily large number of variables depending upon , leading to a contradiction in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). By its nature, it was ineffective (see effective results in number theory); this is of particular interest since a major application of this type of result is to bound the number of solutions of some diophantine equations. Generalizations There is a higher-dimensional version, Schmidt's subspace theorem, of the basic result. There are also numerous extensions, for example using the p-adic metric, based on the Roth method. William J. LeVeque generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field. Define the height H(ξ) of an algebraic number ξ to be the maximum of the absolute values of the coefficients of its minimal polynomial. Fix κ>2. For a given algebraic number α and algebraic number field K, the equation has only finitely many solutions in elements ξ of K. See also Davenport–Schmidt theorem Granville–Langevin conjecture Størmer's theorem Diophantine geometry Notes References Further reading Diophantine approximation Theorems in number theory Algebraic numbers