Binary form

A form in two variables, viz. a homogeneous polynomial

$$ f = f (x, y) = \ \sum _ {k = 0 } ^ { n } a _ {k} x ^ {n - k } y ^ {k} , $$

where the coefficients $ a _ {k} , k = 0 \dots n $ belong to a given commutative ring with a unit element. Such a ring may be the ring $ \mathbf Z $ of integers, the ring of integers of some algebraic number field, the field $ \mathbf R $ of real numbers or the field $ \mathbf C $ of complex numbers. The number $ n $ is called the degree of the form. If $ n = 2, f $ is called a binary quadratic form.

The theory of forms includes algebraic (theory of invariants), arithmetic (representation of numbers by forms) and geometric (theory of arithmetical minima of forms) approaches. The purpose of the algebraic theory of binary forms (in $ \mathbf R $ or $ \mathbf C $) is to construct a complete system of invariants of such forms under linear transformations of variables with coefficients of the same field (cf. Invariants, theory of; see also [2], Chapt. 5). The arithmetic theory of binary forms studies Diophantine equations of the form

$$ f (x, y) = b, $$

where $ a _ {0} \dots a _ {n} , b \in \mathbf Z $, their solvability and their solutions in the ring $ \mathbf Z $. The most important result is Thue's theorem and its generalizations and sharpenings (cf. Thue–Siegel–Roth theorem). See [5], Chapts. 9–17, and the Mordell conjecture on the solvability of such equations in the field $ \mathbf Q $ and the possible number of solutions. The theory of arithmetical minima of binary forms is part of the geometry of numbers. The arithmetical minimum of a form $ f $ is defined as the quantity

$$ m (f) = \inf _ {(x, y) \in \mathbf Z ^ {2} \setminus (0, 0) } \ | f (x, y) | . $$

It has been proved for the case $ n = 3 $ that

$$ m (f) \leq \ \left \{ \begin{array}{ll} | D/49 | ^ {1/4} & \textrm{ if } D > 0, \\ | D/23 | ^ {1/4} & \textrm{ if } D < 0, \\ \end{array} \right .$$

where $ D $ is the discriminant of $ f $, which, in the present case, is

$$ 18a _ {0} a _ {1} a _ {2} a _ {3} + a _ {1} ^ {2} a _ {2} ^ {2} - 4a _ {0} a _ {2} ^ {3} - 4a _ {3} a _ {1} ^ {3} - 27a _ {0} ^ {2} a _ {3} ^ {2} . $$

These estimates cannot be improved.

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2]	G.B. Gurevich, "Foundations of the theory of algebraic invariants" , Noordhoff (1964) (Translated from Russian)
[3]	E. Landau, A. Walfisz, "Diophantische Gleichungen mit endlich vielen Lösungen" , Deutsch. Verlag Wissenschaft. (1959)
[4]	C.G. Lekkerkerker, "Geometry of numbers" , Wolters-Noordhoff (1969)
[5]	L.J. Mordell, "Diophantine equations" , Acad. Press (1969)