Non-homogeneous binary cubic forms

Abstract

1. Let f (x₁, x₂, ..., x_n) be a homogeneous form with real coefficients in n variables x₁, x₂, ..., x_n. Let a₁, a₂, ..., an be n real numbers. Define m_f(a₁, ..., a_n) to be the lower bound of | f(x₁ + a₁, ..., x_n + a_n) | for integers x₁, ..., x_n. Let m_f be the upper bound of m_f(a₁, ..., a_n) for all choices of a₁, ..., a_n. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the region [ƒ(x₁,...x_n)]≤ 1 has a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that: If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that [ƒ(x+a,y+b)]≤ max {[ƒ(½,0)],[ƒ(0,½)],[ƒ(½½)],[ƒ(½,-½)]}.