Binary form

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A form in two variables, viz. a homogeneous polynomial

where the coefficients belong to a given commutative ring with a unit element. Such a ring may be the ring of integers, the ring of integers of some algebraic number field, the field of real numbers or the field of complex numbers. The number is called the degree of the form. If 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 or ) 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

where , their solvability and their solutions in the ring . 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 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 is defined as the quantity

It has been proved for the case that

where is the discriminant of , which, in the present case, is

These estimates cannot be improved.


[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)
How to Cite This Entry:
Binary form. A.V. Malyshev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098