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In [[mathematics]], the '''Jacobian conjecture''' is a celebrated problem on [[polynomial]]s in several [[Variable (mathematics)|variables]]. It was first posed in 1939 by [[Ott-Heinrich Keller]]. It was later named and widely publicised by [[Shreeram Abhyankar]], as an example of a question in the area of [[algebraic geometry]] that requires little beyond a knowledge of [[calculus]] to state. | |||
The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of December 2013, there are no plausible claims to have proved it. | |||
== | ==The Jacobian Determinant== | ||
Let ''N'' > 1 be a fixed integer and consider the polynomials ''f''<sub>1</sub>, ..., ''f<sub>N</sub>'' in variables ''X''<sub>1</sub>, ..., ''X<sub>N</sub>'' with [[coefficient]]s in an [[algebraically closed field]] ''k'' (in fact, it suffices to assume ''k'' = '''[[complex number|C]]'''). Then we define a vector-valued [[function (mathematics)|function]] ''F'': ''k<sup>N</sup>'' → ''k''<sup>''N''</sup> by setting: | |||
: ''F''(''c''<sub>1</sub>, ..., ''c<sub>N</sub>'') = (''f''<sub>1</sub>(''c''<sub>1</sub>, ...,''c''<sub>N</sub>),..., ''f<sub>N</sub>''(''c''<sub>1</sub>,...,''c''<sub>N</sub>)) | |||
The [[Jacobian matrix and determinant|Jacobian determinant]] of ''F'', denoted by ''J<sub>F</sub>'', is defined as the [[determinant]] of the ''N'' × ''N'' [[matrix (mathematics)|matrix]] consisting of the [[partial derivative]]s of ''f<sub>i</sub>'' with respect to ''X<sub>j</sub>'': | |||
:<math>J_F = \left | \begin{matrix} \frac{\partial f_1}{\partial X_1} & \cdots & \frac{\partial f_1}{\partial X_N} \\ | |||
\vdots & \ddots & \vdots \\ | |||
\frac{\partial f_N}{\partial X_1} & \cdots & \frac{\partial f_N}{\partial X_N} \end{matrix} \right |,</math> | |||
then ''J<sub>F</sub>'' is itself a polynomial function of the ''N'' variables ''X''<sub>1</sub>, …, ''X<sub>N</sub>''. | |||
==Formulation of the Conjecture== | |||
The condition ''J<sub>F</sub>'' ≠ 0 is related to the [[inverse function theorem]] in [[multivariable calculus]]. In fact for smooth functions (and so in particular for polynomials) a local inverse function to ''F'' exists at any point where ''J<sub>F</sub>'' is non-zero. However ''k'' is algebraically closed so ''J<sub>F</sub>'' as a polynomial will be zero for some complex values of ''X''<sub>1</sub>, …, ''X<sub>N</sub>'' unless it is a non-zero [[constant function]]. It is a relatively elementary fact that: | |||
<blockquote>'''Proposition:''' If ''F'' has an [[inverse function]] ''G'': ''k<sup>N</sup>'' → ''k<sup>N</sup>'', then ''J<sub>F</sub>'' is a non-zero constant.</blockquote> | |||
The conjecture is the following converse: | |||
<blockquote>'''Jacobian conjecture:''' If ''J<sub>F</sub>'' is a non-zero constant, then ''F'' has an inverse function ''G'': ''k<sup>N</sup>'' → ''k<sup>N</sup>'', and ''G'' is [[regular map (algebraic geometry)|regular]] (in the sense that its components are given by polynomial expressions).</blockquote> | |||
==Results== | |||
{{harvtxt|Wang|1980}} proved the Jacobian conjecture for polynomials of [[degree of a polynomial|degree]] 2, and {{harvtxt|Bass|Connell|Wright|1982}} showed that the general case follows from the special case where the polynomials are of degree 3. {{harvtxt|Moh|1983}} checked the conjecture for polynomials of degree at most 100 in 2 variables. | |||
The Jacobian conjecture is equivalent to the [[Dixmier conjecture]]. | |||
==See also== | |||
*[[Smale's problems]] | |||
*[[Dixmier conjecture]] | |||
==Notes== | |||
<references/> | |||
==References== | |||
*{{Citation | last1=Bass | first1=Hyman | last2=Connell | first2=Edwin H. | last3=Wright | first3=David | title=The Jacobian conjecture: reduction of degree and formal expansion of the inverse | doi=10.1090/S0273-0979-1982-15032-7 | isbn=979-1982150327 {{Please check ISBN|reason=Check digit (7) does not correspond to calculated figure.}}{{Please check ISBN|reason=does not match hyphenation rules and is probably invalid}} | mr=663785 | year=1982 | journal=American Mathematical Society. Bulletin. New Series | volume=7 | issue=2 | pages=287–330}} | |||
*{{Citation | last1=Belov-Kanel | first1=Alexei | last2=Kontsevich | first2=Maxim | title=The Jacobian conjecture is stably equivalent to the Dixmier conjecture | arxiv=math/0512171 | mr=2337879 | year=2007 | journal=Moscow Mathematical Journal | volume=7 | issue=2 | pages=209–218 | bibcode=2005math.....12171B}} | |||
*{{springer|id=J/j120010|title=Jacobian conjecture|author=A. van den Essen}} | |||
*{{Citation | last1=Keller | first1=Ott-Heinrich | title=Ganze Cremona-Transformationen | url=http://dx.doi.org/10.1007/BF01695502 | doi=10.1007/BF01695502 | year=1939 | journal=Monatshefte für Mathematik und Physik | issn=0026-9255 | volume=47 | issue=1 | pages=299–306}} | |||
*{{Citation | last1=Moh | first1=T. T. | title=On the Jacobian conjecture and the configurations of roots | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002200376 | id={{MR|691964}} Preprint titled "On the global Jacobian conjecture for polynomials of degree less than 100" | year=1983 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=340 | pages=140–212}} | |||
*A. van den Essen, ''Polynomial automorphisms and the Jacobian conjecture'', ISBN 3-7643-6350-9 (http://emis.mi.ras.ru/journals/SC/1997/2/pdf/smf_sem-cong_2_55-81.pdf). | |||
*{{Citation | last1=Wang | first1=Stuart Sui-Sheng | title=A Jacobian criterion for separability | journal=Journal of Algebra | volume=65 | pages=453–494 | url=http://www.sciencedirect.com/science/journal/00218693/65/2}} | |||
==External links== | |||
*[http://www.math.purdue.edu/~ttm/jacobian.html Web page of T. T. Moh on the conjecture] | |||
{{DEFAULTSORT:Jacobian Conjecture}} | |||
[[Category:Polynomials]] | |||
[[Category:Algebraic geometry]] | |||
[[Category:Conjectures]] | |||
Revision as of 16:42, 12 May 2013
In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was later named and widely publicised by Shreeram Abhyankar, as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state.
The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of December 2013, there are no plausible claims to have proved it.
The Jacobian Determinant
Let N > 1 be a fixed integer and consider the polynomials f1, ..., fN in variables X1, ..., XN with coefficients in an algebraically closed field k (in fact, it suffices to assume k = C). Then we define a vector-valued function F: kN → kN by setting:
- F(c1, ..., cN) = (f1(c1, ...,cN),..., fN(c1,...,cN))
The Jacobian determinant of F, denoted by JF, is defined as the determinant of the N × N matrix consisting of the partial derivatives of fi with respect to Xj:
then JF is itself a polynomial function of the N variables X1, …, XN.
Formulation of the Conjecture
The condition JF ≠ 0 is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a local inverse function to F exists at any point where JF is non-zero. However k is algebraically closed so JF as a polynomial will be zero for some complex values of X1, …, XN unless it is a non-zero constant function. It is a relatively elementary fact that:
Proposition: If F has an inverse function G: kN → kN, then JF is a non-zero constant.
The conjecture is the following converse:
Jacobian conjecture: If JF is a non-zero constant, then F has an inverse function G: kN → kN, and G is regular (in the sense that its components are given by polynomial expressions).
Results
Template:Harvtxt proved the Jacobian conjecture for polynomials of degree 2, and Template:Harvtxt showed that the general case follows from the special case where the polynomials are of degree 3. Template:Harvtxt checked the conjecture for polynomials of degree at most 100 in 2 variables.
The Jacobian conjecture is equivalent to the Dixmier conjecture.
See also
Notes
References
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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
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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 - Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
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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
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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 - A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, ISBN 3-7643-6350-9 (http://emis.mi.ras.ru/journals/SC/1997/2/pdf/smf_sem-cong_2_55-81.pdf).
- 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