Kodaira dimension: Difference between revisions
en>RedBot m r2.5.2) (Robot: Modifying sl:Kodairova razsežnost |
The power n notation is used in this article for tensor powers later, to avoid confusion clarifying that canonical bundle is an exterior power. |
||
| Line 1: | Line 1: | ||
{{more footnotes|date=July 2009}} | |||
In [[mathematics]], a '''binary quadratic form''' is a [[quadratic form]] in two variables. More concretely, it is a [[homogeneous polynomial]] of degree 2 in two variables | |||
: <math> q(x,y)=ax^2+bxy+cy^2, \, </math> | |||
where ''a'', ''b'', ''c'' are the '''coefficients'''. Properties of binary quadratic forms depend in an essential way on the nature of the coefficients, which may be [[real number]]s, [[rational number]]s, or in the most delicate case, [[integer]]s. Arithmetical aspects of the theory of binary quadratic forms are related to the arithmetic of [[quadratic field]]s and have been much studied, notably, by [[Gauss]] in Section V of ''[[Disquisitiones Arithmeticae]]''. The theory of binary quadratic forms has been extended in two directions: general [[number field]]s and quadratic forms in ''n'' variables. | |||
==Brief history== | |||
Binary quadratic forms were considered already by [[Fermat]], in particular, in the question of [[Fermat's theorem on sums of two squares|representations of numbers as sums of two squares]]. The theory of [[Pell's equation]] may be viewed as a part of the theory of binary quadratic forms. [[Lagrange]] in 1773 initiated the development of the general theory of quadratic forms. First systematic treatment of binary quadratic forms is due to [[Adrien-Marie Legendre|Legendre]]. Their theory was advanced much further by [[Carl Friedrich Gauss|Gauss]] in ''[[Disquisitiones Arithmeticae]]''. He considered questions of equivalence and reduction and introduced composition of binary quadratic forms (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. | |||
==Main questions== | |||
A classical question in the theory of integral quadratic forms (those with integer coefficients) is the '''representation problem''': describe the set of numbers represented by a given quadratic form ''q''. If the number of representations is finite then a further question is to give a closed formula for this number. The notion of ''equivalence'' of quadratic forms and the related ''reduction theory'' are the principal tools in addressing these questions. | |||
Two integral forms are called '''equivalent''' if there exists an invertible integral linear change of variables that transforms the first form into the second. This defines an [[equivalence relation]] on the set of integral quadratic forms, whose elements are called '''classes''' of quadratic forms. Equivalent forms necessarily have the same [[discriminant of a quadratic form|discriminant]] | |||
: <math> D(f)=b^2-4ac, \quad D(f)\equiv 0,1\, (\!\!\!\!\! \mod 4). </math> | |||
Gauss proved that for every value ''D'', there are only finitely many classes of binary quadratic forms with discriminant ''D''. Their number is the '''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. One of the deepest discoveries of Gauss was the existence of a natural '''composition law''' on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite [[abelian group]] called the '''class group''' of discriminant ''D''. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each '''[[Genus of a quadratic form|genus]]''' consists of finitely many classes of forms. | |||
An integral binary quadratic form is called '''primitive''' if ''a'', ''b'', and ''c'' have no common factor. If a form's discriminant is a [[fundamental discriminant]], then the form is primitive.<ref>{{harvnb|Cohen|1993|loc=§5.2}}</ref> | |||
From a modern perspective, the class group of a fundamental discriminant ''D'' is [[isomorphic]] to the [[narrow class group]] of the [[quadratic field]] <math>\mathbf{Q}(\sqrt{D})</math> of discriminant ''D''.<ref>{{harvnb|Fröhlich|Taylor|1993|loc=Theorem 58}}</ref> For negative ''D'', the narrow class group is the same as the [[ideal class group]], but for positive ''D'' it may be twice as big. | |||
==See also== | |||
* [[Legendre symbol]] | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
* Johannes Buchmann, Ulrich Vollmer: ''Binary Quadratic Forms'', Springer, Berlin 2007, ISBN 3-540-46367-4 | |||
* Duncan A. Buell: ''Binary Quadratic Forms'', Springer, New York 1989 | |||
* {{Citation | |||
| last=Cohen | |||
| first=Henri | |||
| author-link=Henri Cohen (number theorist) | |||
| title=A Course in Computational Algebraic Number Theory | |||
| publisher=[[Springer-Verlag]] | |||
| location=Berlin, New York | |||
| series=Graduate Texts in Mathematics | |||
| isbn=978-3-540-55640-4 | |||
| mr=1228206 | |||
| year=1993 | |||
| volume=138 | |||
}} | |||
* {{Citation | |||
| last= Fröhlich | |||
| first = Albrecht | |||
| authorlink= Albrecht Fröhlich | |||
| last2=Taylor | |||
| first2=Martin | |||
| authorlink2= Martin J. Taylor | |||
| title=Algebraic number theory | |||
| publisher=[[Cambridge University Press]] | |||
| series=Cambridge Studies in Advanced Mathematics | |||
| isbn=978-0-521-43834-6 | |||
| mr=1215934 | |||
| year=1993 | |||
| volume=27 | |||
}} | |||
==External links== | |||
* {{eom|id=b/b016370|author=A.V.Malyshev|title=Binary quadratic form}} | |||
[[Category:Quadratic forms]] | |||
Revision as of 13:16, 30 January 2014
Template:More footnotes In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variables
where a, b, c are the coefficients. Properties of binary quadratic forms depend in an essential way on the nature of the coefficients, which may be real numbers, rational numbers, or in the most delicate case, integers. Arithmetical aspects of the theory of binary quadratic forms are related to the arithmetic of quadratic fields and have been much studied, notably, by Gauss in Section V of Disquisitiones Arithmeticae. The theory of binary quadratic forms has been extended in two directions: general number fields and quadratic forms in n variables.
Brief history
Binary quadratic forms were considered already by Fermat, in particular, in the question of representations of numbers as sums of two squares. The theory of Pell's equation may be viewed as a part of the theory of binary quadratic forms. Lagrange in 1773 initiated the development of the general theory of quadratic forms. First systematic treatment of binary quadratic forms is due to Legendre. Their theory was advanced much further by Gauss in Disquisitiones Arithmeticae. He considered questions of equivalence and reduction and introduced composition of binary quadratic forms (Gauss and many subsequent authors wrote 2b in place of b; the modern convention allowing the coefficient of xy to be odd is due to 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 fields.
Main questions
A classical question in the theory of integral quadratic forms (those with integer coefficients) is the representation problem: describe the set of numbers represented by a given quadratic form q. If the number of representations is finite then a further question is to give a closed formula for this number. The notion of equivalence of quadratic forms and the related reduction theory are the principal tools in addressing these questions.
Two integral forms are called equivalent if there exists an invertible integral linear change of variables that transforms the first form into the second. This defines an equivalence relation on the set of integral quadratic forms, whose elements are called classes of quadratic forms. Equivalent forms necessarily have the same discriminant
Gauss proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the 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. One of the deepest discoveries of Gauss was the existence of a natural composition law on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite abelian group called the class group of discriminant D. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each genus consists of finitely many classes of forms.
An integral binary quadratic form is called primitive if a, b, and c have no common factor. If a form's discriminant is a fundamental discriminant, then the form is primitive.[1]
From a modern perspective, the class group of a fundamental discriminant D is isomorphic to the narrow class group of the quadratic field of discriminant D.[2] For negative D, the narrow class group is the same as the ideal class group, but for positive D it may be twice as big.
See also
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- Johannes Buchmann, Ulrich Vollmer: Binary Quadratic Forms, Springer, Berlin 2007, ISBN 3-540-46367-4
- Duncan A. Buell: Binary Quadratic Forms, Springer, New York 1989
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010