View source for Binary form

A form in two variables, viz. a homogeneous [[Polynomial|polynomial]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163401.png" /></td> </tr></table>

where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163402.png" /> belong to a given [[Commutative ring|commutative ring]] with a unit element. Such a ring may be the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163403.png" /> of integers, the ring of integers of some algebraic number field, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163404.png" /> of real numbers or the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163405.png" /> of complex numbers. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163406.png" /> is called the degree of the form. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163407.png" /> is called a [[Binary quadratic form|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163408.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b0163409.png" />) 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|Invariants, theory of]]; see also [[#References|[2]]], Chapt. 5). The arithmetic theory of binary forms studies Diophantine equations of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634011.png" />, their solvability and their solutions in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634012.png" />. The most important result is Thue's theorem and its generalizations and sharpenings (cf. [[Thue–Siegel–Roth theorem|Thue–Siegel–Roth theorem]]). See [[#References|[5]]], Chapts. 9–17, and the [[Mordell conjecture|Mordell conjecture]] on the solvability of such equations in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634013.png" /> and the possible number of solutions. The theory of arithmetical minima of binary forms is part of the [[Geometry of numbers|geometry of numbers]]. The arithmetical minimum of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634014.png" /> is defined as the quantity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634015.png" /></td> </tr></table>

It has been proved for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634016.png" /> that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634018.png" /> is the discriminant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634019.png" />, which, in the present case, is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016340/b01634020.png" /></td> </tr></table>

These estimates cannot be improved.

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