An isolation theorem for decomposable forms of purely real algebraic fields of degree $n\ge3$

B. F. Skubenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 4, Tome 112 (1981), pp. 167-171

Résumé

Let $M$ be a complete module of a purely algebraic field of degree $n\ge3$, let $\Lambda$ be the lattice of this module and let $F(X)$ be its form. By $\Lambda_\varepsilon$ we denote any lattice for which we have $\Lambda_\varepsilon=\tau\Lambda$, where $\tau$ is a nondiagonal matrix satisfying the condition $\|\tau-I\|\le\varepsilon$, $I$ being the identity matrix. The complete collection of such lattices will be denoted by $\{\Lambda_\varepsilon\}$. To each lattice $\Lambda_\varepsilon$ we associate in a natural manner the decomposable form $F_\varepsilon(X)$. The complete collection of forms, corresponding to the set $\{\Lambda_\varepsilon\}$, will be denoted by $\{F_\varepsilon\}$. It is shown that for any given arbitrarily small interval $(N-\eta,N+\eta)$, one can select an $\varepsilon$ one can select an $F_\varepsilon(X)$ from $\{F_\varepsilon\}$ there exists an integral vector $X_0$ such that $N-\eta<F_\varepsilon(X_0)<N+\eta$.