Non-homogeneous binary cubic forms

Bambah, R. P. (1951) Non-homogeneous binary cubic forms Mathematical Proceedings of the Cambridge Philosophical Society, 47 (3). pp. 457-460. ISSN 0305-0041

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1. Let f (x1, x2, ..., xn) be a homogeneous form with real coefficients in n variables x1, x2, ..., xn. Let a1, a2, ..., an be n real numbers. Define mf(a1, ..., an) to be the lower bound of | f(x1 + a1, ..., xn + an) | for integers x1, ..., xn. Let mf be the upper bound of mf(a1, ..., an) for all choices of a1, ..., an. 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,...xn)]≤ 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)]}.

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