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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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S0305004100026840

## Abstract

1. Let f (x_{1}, x_{2}, ..., x_{n}) be a homogeneous form with real coefficients in n variables x_{1}, x_{2}, ..., x_{n}. Let a_{1}, a_{2}, ..., an be n real numbers. Define m_{f}(a_{1}, ..., a_{n}) to be the lower bound of | f(x_{1} + a_{1}, ..., x_{n} + a_{n}) | for integers x_{1}, ..., x_{n}. Let m_{f} be the upper bound of m_{f}(a_{1}, ..., a_{n}) for all choices of a_{1}, ..., 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(x_{1},...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)]}.

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ID Code: | 71969 |

Deposited On: | 28 Nov 2011 04:52 |

Last Modified: | 28 Nov 2011 04:52 |

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