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{{Short description|Quadratic homogeneous polynomial in two variables}}
{{About|binary quadratic forms with [[integer]] coefficients|binary quadratic forms with other coefficients|quadratic form}}
{{more footnotes|date=July 2009}}
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where ''a'', ''b'', ''c'' are the '''coefficients'''. When the coefficients can be arbitrary [[complex number]]s, most results are not specific to the case of two variables, so they are described in [[quadratic form]]. A quadratic form with [[integer]] coefficients is called an '''integral binary quadratic form''', often abbreviated to ''binary quadratic form''.
This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of [[algebraic number theory]]. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to [[quadratic field|quadratic]] and more general [[number field]]s, but advances specific to binary quadratic forms still occur on occasion.
Pierre Fermat stated that if p is an odd prime then the equation <math>p = x^2 + y^2</math> has a solution iff <math>p \equiv 1 \pmod{4}</math>, and he made similar statement about the equations <math>p = x^2 + 2y^2</math>, <math>p = x^2 + 3y^2</math>, <math>p = x^2 - 2y^2</math> and <math>p = x^2 - 3y^2</math>.
<math>x^2 + y^2, x^2 + 2y^2, x^2 - 3y^2</math> and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems.
Another instance of quadratic forms is [[Pell's equation]] <math>x^2-ny^2=1</math>.
Binary quadratic forms are closely related to ideals in quadratic fields. This allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant.
The classical [[theta function]] of 2 variables is <math> \sum_{(m,n)\in \mathbb{Z}^2} q^{m^2 + n^2}</math>, if <math>f(x,y)</math> is a positive [[definite quadratic form]] then <math> \sum_{(m,n)\in \mathbb{Z}^2} q^{f(m,n)}</math> is a theta function.
== Equivalence ==
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: <math>\begin{align} f(\alpha x + \beta y, \gamma x + \delta y) &= g(x,y),\\
\alpha \delta - \beta \gamma &= 1.\end{align}
For example, with <math>f= x^2 + 4xy + 2y^2</math> and <math>\alpha = -3</math>, <math>\beta = 2</math>, <math>\gamma = 1</math>, and <math>\delta = -1</math>, we find that ''f'' is equivalent to <math>g = (-3x+2y)^2 + 4(-3x+2y)(x-y)+2(x-y)^2</math>, which simplifies to <math>-x^2+4xy-2y^2</math>.
The above equivalence conditions define an [[equivalence relation]] on the set of integral quadratic forms. It follows that the quadratic forms are [[partition of a set|partition]]ed
Lagrange used a different notion of equivalence, in which the second condition is replaced by <math> \alpha \delta - \beta \gamma = \pm 1</math>. Since Gauss it has been recognized that this definition is inferior to that given above. If there is a need to distinguish, sometimes forms are called '''properly equivalent''' using the definition above and '''improperly equivalent''' if they are equivalent in Lagrange's sense.
In [[matrix (mathematics)|matrix]] terminology, which is used occasionally below, when
: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
has integer entries and determinant 1, the map <math> f(x,y) \mapsto f(\alpha x + \beta y, \gamma x + \delta y)</math> is a (right) [[Group action (mathematics)|group action]] of <math>\mathrm{SL}_2(\mathbb{Z})</math> on the set of binary quadratic forms. The equivalence relation above then arises from the general theory of group actions.
If <math>f=ax^2+bxy+cy^2</math>, then important invariants include
* The [[discriminant]] <math>\Delta=b^2-4ac
* The content, equal to the greatest common divisor of ''a'', ''b'', and ''c''.
Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant <math>\Delta</math> is '''definite''' if <math>\Delta < 0</math>, '''degenerate''' if <math>\Delta</math> is a perfect square, and '''indefinite''' otherwise. A form is '''primitive''' if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a [[fundamental discriminant]], then the form is primitive.<ref>{{harvnb|Cohen|1993|loc=§5.2}}</ref> Discriminants satisfy <math>\Delta\equiv 0,1 \pmod 4. </math>
=== Automorphisms ===
If ''f'' is a quadratic form, a matrix
: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
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: <math> \begin{pmatrix} 3 & -4 \\ -2 & 3 \end{pmatrix} </math>
is an automorphism of the form <math>f = x^2 - 2y^2</math>. The automorphisms of a form
==
=== Examples ===
[[Diophantus]] considered whether, for an odd integer <math>n</math>, it is possible to find integers <math>x</math> and <math>y</math> for which <math>n = x^2 + y^2</math>.<ref>{{harvnb|Weil|2001|p
: <math>\begin{align} 65 &= 1^2 + 8^2,\\
65 &= 4^2 + 7^2,
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: <math>\begin{align}
(3 \cdot 17 + 4 \cdot 12, 2 \cdot 17 + 3 \cdot 12) &= (99,70),\\
(3 \cdot 99 + 4 \cdot 70, 2 \cdot 99 + 3 \cdot 70) &= (
&\vdots \end{align}
</math>
These values will keep growing in size, so we see there are infinitely many ways to represent 1 by the form <math>x^2 - 2y^2</math>. This recursive description was discussed in Theon of Smyrna's commentary on [[Euclid's Elements]].
=== The representation problem ===
The oldest problem in the theory of binary quadratic forms is the '''representation problem''': describe the representations of a given number <math>n</math> by a given quadratic form ''f''. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of
The examples above discuss the representation problem for the numbers 3 and 65 by the form <math>x^2 + y^2</math> and for the number 1 by the form <math>x^2 - 2y^2</math>. We see that 65 is represented by <math>x^2 + y^2</math> in sixteen different ways, while 1 is represented by <math>x^2 - 2y^2</math> in infinitely many ways and
3 is not represented by <math>x^2+y^2</math> at all. In the first case, the sixteen representations were explicitly described. It was also shown that the number of representations of an integer by <math>x^2+y^2</math> is always finite. The [[sum of squares function]] <math>r_2(n)</math> gives the number of representations of ''n'' by <math>x^2+y^2</math> as a function of ''n''. There is a closed formula<ref>{{harvnb|Hardy|Wright|2008
: <math> r_2(n) = 4(d_1(n) - d_3(n)), </math>
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The minimum absolute value represented by a class is zero for degenerate classes and positive for definite and indefinite classes. All numbers represented by a definite form <math>f = ax^2 + bxy + cy^2</math> have the same sign: positive if <math>a>0</math> and negative if <math>a<0</math>. For this reason, the former are called '''positive definite''' forms and the latter are '''negative definite'''.
The number of representations of an integer ''n'' by a form ''f'' is finite if ''f'' is definite and infinite if ''f'' is indefinite. We saw instances of this in the examples above: <math>x^2+y^2</math> is positive definite and <math>x^2 - 2y^2</math> is indefinite.
=== Equivalent representations ===
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: <math>\begin{pmatrix} \delta& -\beta \\ -\gamma & \alpha\end{pmatrix} \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}</math>
The above conditions give a (right) action of the group <math>\mathrm{SL}_2(\mathbb{Z})</math> on the set of representations of integers by binary quadratic forms. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.
As an example, let <math>f = x^2 - 2y^2</math> and consider a representation <math>1 = f(x_1,y_1)</math>. Such a representation is a solution to the Pell equation described in the examples above. The matrix
: <math> \begin{pmatrix} 3 & -4 \\ -2 & 3 \end{pmatrix} </math>
has determinant 1 and is an automorphism of ''f''. Acting on the representation <math>1 = f(x_1,y_1)</math> by this matrix yields the equivalent representation <math>1 = f(3x_1 + 4y_1, 2x_1 + 3 y_1)</math>. This is the recursion step in the process described above for generating infinitely many solutions to <math>1 = x^2 - 2y^2</math>. Iterating this matrix action, we find that the infinite set of representations of 1 by ''f'' that were determined above are all equivalent.
There are generally finitely many equivalence classes of representations of an integer ''n'' by forms of given nonzero discriminant <math>\Delta</math>. A complete set of
== Reduction and class numbers<!--'Class number (binary quadratic forms)' redirects here--> ==
Lagrange proved that for every value ''D'', there are only finitely many classes of binary quadratic forms with discriminant ''D''. Their number is the '''{{vanchor|class number}}'''<!--boldface per WP:R#PLA--> of discriminant ''D''. He described an algorithm, called '''reduction''', for constructing a canonical representative in each class, the '''reduced form''', whose coefficients are the smallest in a suitable sense.▼
Gauss gave a superior reduction algorithm in ''[[Disquisitiones Arithmeticae]]'', which
▲Lagrange proved that for every value ''D'', there are only finitely many classes of binary quadratic forms with discriminant ''D''. Their number is the '''{{vanchor|class number}}''' of discriminant ''D''. He described an algorithm, called '''reduction''', for constructing a canonical representative in each class, the '''reduced form''', whose coefficients are the smallest in a suitable sense.
▲Gauss gave a superior reduction algorithm in ''[[Disquisitiones Arithmeticae]]'', which has ever since the reduction algorithm most commonly given in textbooks. In 1981, Zagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss's.<ref>{{harvnb|Zagier|1981||loc=}}</ref>
== Composition ==
'''Composition''' most commonly refers to a [[binary operation]] on primitive equivalence classes of forms of the same discriminant
"Composition" also sometimes refers to, roughly, a binary operation on binary quadratic forms. The word "roughly" indicates two caveats: only certain pairs of binary quadratic forms can be composed, and the resulting form is not well-defined (although its equivalence class is). The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes.
"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated{{citation needed|date=March 2017}}<!--<ref>{{harvnb|Shanks|1989}}</ref> full citation not in article yet --> than composition of forms, but arose first historically. We will consider such operations in a separate section below.
Composition means taking 2 quadratic forms of the same discriminant and combining them to create a quadratic form of the same discriminant, as follows from [[Brahmagupta's identity]].
=== Composing forms and classes ===
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A variety of definitions of composition of forms has been given, often in an attempt to simplify the extremely technical and general definition of Gauss. We present here Arndt's method, because it remains rather general while being simple enough to be amenable to computations by hand. An alternative definition is described at [[Bhargava cube]]s.
Suppose we wish to compose forms <math>f_1 =
# Compute <math>
# Solve the system of congruences <blockquote><math>\begin{align} x &\equiv B_1 \pmod{2 \tfrac{A_1}{e}}\\ x &\equiv B_2 \pmod{2 \tfrac{A_2}{e}}\\ \tfrac{
# Compute ''C'' such that <math>\Delta = B^2 - 4AC</math>. It can be shown that ''C'' is an integer.
The form <math>Ax^2 + Bxy + Cy^2</math> is "the" composition of <math>f_1</math> and <math>f_2</math>. We see that its first coefficient is well-defined, but the other two depend on the choice of ''B'' and ''C''. One way to make this a well-defined operation is to make an arbitrary convention for how to choose ''B''
: <math>\begin{pmatrix} 1 & n\\ 0 & 1\end{pmatrix}</math>,
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== History ==
There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms.<ref>{{harvnb|Weil|2001|loc=Ch.I §§VI, VIII}}</ref> The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of [[Pell's equation]] and the representation of integers as
The general theory of quadratic forms was initiated by [[Lagrange]] in 1775 in his ''[[List of important publications in mathematics#Recherches d'Arithmétique|Recherches d'Arithmétique]]''. Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms."<ref>{{harvnb|Weil|2001|loc=p.318}}</ref> He was the first to recognize the importance of the discriminant and to define the essential notions of equivalence and reduction, which, according to Weil, have "dominated the whole subject of quadratic forms ever since".<ref>{{harvnb|Weil|2001|loc=p.317}}</ref> Lagrange showed that there are finitely many equivalence classes of given discriminant, thereby defining for the first time an arithmetic [[Ideal class group|class number]]. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of [[infrastructure (number theory)|infrastructure]]. In 1798, [[Adrien-Marie Legendre|Legendre]] published ''Essai sur la théorie des nombres'', which summarized the work of Euler and Lagrange and added some of his own contributions, including the first glimpse of a composition operation on forms.
The theory was vastly extended and refined by [[Carl Friedrich Gauss|Gauss]] in Section V of ''[[List of important publications in mathematics#Disquisitiones Arithmeticae|Disquisitiones Arithmeticae]]''. Gauss introduced a very general version of a [[composition operator]] that allows composing even forms of different discriminants and imprimitive forms. He replaced Lagrange's equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a [[group (mathematics)|group]] under the composition operation. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. (Gauss and many subsequent authors wrote 2''b'' in place of ''b''; the modern convention allowing the coefficient of ''xy'' to be odd is due to [[Gotthold Eisenstein|Eisenstein]]).
These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general [[number field]]s. But the impact was not immediate. Section V of ''Disquisitiones'' contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader. Combined, the novelty and complexity made Section V notoriously difficult. [[Dirichlet]] published simplifications of the theory that made it accessible to a broader audience. The culmination of this work is his text ''[[List of important publications in mathematics#Vorlesungen über Zahlentheorie|Vorlesungen über Zahlentheorie]]''. The third edition of this work includes two supplements by [[Dedekind]]. Supplement XI introduces [[ring theory]], and from then on, especially after the 1897 publication of [[Hilbert|Hilbert's]] ''[[List of important publications in mathematics#Zahlbericht|Zahlbericht]]'', the theory of binary quadratic forms lost its preeminent position in [[algebraic number theory]] and became overshadowed by the more general theory of [[algebraic number fields]].
Even so, work on binary quadratic forms with integer coefficients continues to the present. This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including [[Daniel Shanks|
==See also==
* [[Bhargava cube]]
*[[Fermat's theorem on sums of two squares]]
* [[Legendre symbol]]
* [[Brahmagupta's identity]]
==Notes==
{{reflist|30em}}
==References==
* Johannes Buchmann, Ulrich Vollmer: ''Binary Quadratic Forms'', Springer, Berlin 2007, {{ISBN
* Duncan A. Buell: ''Binary Quadratic Forms'', Springer, New York 1989
* David A. Cox, ''Primes of the form <math>x^2 + ny^2</math>, Fermat, class field theory, and complex multiplication''
* {{Citation
| last=Cohen
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| last= Fröhlich
| first = Albrecht
|
| last2=Taylor
| first2=Martin
|
| title=Algebraic number theory
| publisher=[[Cambridge University Press]]
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| volume=27
}}
* {{
| last1=Hardy
| first1=G. H.
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| first2=E. M.
| author2-link=E. M. Wright
| edition=
| others=
| title=An Introduction to the Theory of Numbers
| publisher=
| ___location=Oxford
| isbn= 978-0-19-921986-5
| mr= 2445243
| zbl= 1159.11001
| year=2008
▲| origyear=1938
}}
* {{Citation
| last = Weil
| first = André
|
| title = Number Theory: An approach through history from Hammurapi to Legendre
| publisher = Birkhäuser Boston
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| last = Zagier
| first = Don
|
| title = Zetafunktionen und quadratische Körper: eine Einführung in die höhere Zahlentheorie
| publisher = Springer
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==External links==
* [http://oeis.org/wiki/User:Peter_Luschny/BinaryQuadraticForms Peter Luschny, Positive numbers represented by a binary quadratic form]
* {{eom|id=b/b016370|author=A. V. Malyshev|title=Binary quadratic form}}
[[Category:Quadratic forms]]
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