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In [[mathematics]], '''Hilbert's fourteenth problem''', that is, number 14 of [[Hilbert's problems]] proposed in 1900, asks whether certain [[ring (mathematics)|ring]]s are [[finitely generated algebra|finitely generated]]. | |||
The setting is as follows: Assume that ''k'' is a [[field (mathematics)|field]] and let ''K'' be a subfield of the field of [[rational function]]s in ''n'' variables, | |||
:''k''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> ) over ''k''. | |||
Consider now the ring ''R'' defined as the intersection | |||
= | :<math> R:= K \cap k[x_1, \dots, x_n] \ .</math> | ||
Hilbert conjectured that all such subrings are finitely generated. It can be shown that the field ''K'' is always finitely generated ''as a field'', in other words, there exist finitely many elements | |||
= | :''y''<sub>''i''</sub>, ''i'' = 1 ,...,''d'' in ''K'' | ||
such that every element in ''R'' can be ''rationally'' represented by the ''y''<sub>''i''</sub>. But this does not imply that the ring ''R'' is finitely generated ''as a ring'', even if all the elements ''y''<sub>i</sub> could be chosen from ''R''. | |||
After some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for ''n'' = 1 and ''n'' = 2 by [[Zariski]] in 1954) then in 1959 [[Masayoshi Nagata]] found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a [[linear algebraic group]]. | |||
== History == | |||
The problem originally arose in algebraic [[invariant theory]]. Here the ring ''R'' is given as a (suitably defined) ring of polynomial invariants of a [[linear algebraic group]] over a field ''k'' acting algebraically on a [[polynomial ring]] ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] (or more generally, on a finitely generated algebra defined over a field). In this situation the field ''K'' is the field of ''rational'' functions (quotients of polynomials) in the variables ''x''<sub>''i''</sub> which are invariant under the given action of the algebraic group, the ring ''R'' is the ring of ''polynomials'' which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by [[Arthur Cayley|Cayley]], [[James Joseph Sylvester|Sylvester]], [[Alfred Clebsch|Clebsch]], [[Paul Gordan]] and also Hilbert) of invariants of [[binary quantic|binary form]]s in two variables with the natural action of the [[special linear group]] ''SL''<sub>2</sub>(''k'') on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of [[complex number]]s for some classical [[semi-simple]] [[Lie group]]s (in particular the [[general linear group]] over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. This finiteness result was later extended by [[Hermann Weyl]] to the class of all semi-simple Lie-groups. A major ingredient in Hilbert's proof is the [[Hilbert basis theorem]] applied to the [[ideal (ring theory)|ideal]] inside the polynomial ring generated by the invariants. | |||
== Zariski's formulation == | |||
[[Oscar Zariski|Zariski]]'s formulation of Hilbert's fourteenth problem asks whether, for a [[Quasiprojective variety|quasi-affine]] [[algebraic variety]] ''X'' over a field ''k'', possibly assuming ''X'' [[Normal scheme|normal]] or [[Smooth (algebraic variety)|smooth]], the ring of [[regular function]]s on ''X'' is finitely generated over ''k''. | |||
Zariski's formulation was shown<ref>{{Citation | last = Winkelmann | first = Jörg | title = Invariant rings and quasiaffine quotients | journal = Math. Z. | volume = 244 | issue = 1 | pages = 163–174 | year = 2003 | doi = 10.1007/s00209-002-0484-9 | postscript = .}}</ref> to be equivalent to the original problem, for ''X'' normal. | |||
==Nagata's counterexample== | |||
{{harvtxt|Nagata|1958}} gave the following counterexample to Hilbert's problem. The field ''k'' is a field containing 48 elements ''a''<sub>1''i''</sub>, ...,''a''<sub>16''i''</sub>, for ''i''=1, 2, 3 that are algebraically independent over the prime field. The ring ''R'' is the polynomial ring ''k''[''x''<sub>1</sub>,...,''x''<sub>16</sub>, ''t''<sub>1</sub>,...,''t''<sub>16</sub>] in 32 variables. The vector space ''V'' is a 13-dimensional vector space over ''k'' consisting of all vectors (''b''<sub>1</sub>,...,''b''<sub>16</sub>) in ''k''<sup>16</sup> orthogonal to each of the three vectors (''a''<sub>1''i''</sub>, ...,''a''<sub>16''i''</sub>) for ''i''=1, 2, 3. The vector space ''V'' is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on ''R'' by fixing all elements ''t''<sub>''j''</sub> and taking ''x''<sub>''j''</sub> to ''x''<sub>''j''</sub> + ''b''<sub>''j''</sub>''t''<sub>''j''</sub>. Then the ring of elements of ''R'' invariant under the action of the group ''V'' is not a finitely generated ''k''-algebra. | |||
Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example, {{harvtxt|Totaro|2008}} showed that over any field there is an action of the sum ''G''{{su|p=3|b=a}} of three copies of the additive group on ''k''<sup>18</sup> whose ring of invariants is not finitely generated. | |||
== References == | |||
*{{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Proc. Internat. Congress Math. 1958 | url=http://mathunion.org/ICM/ICM1958/ | publisher=[[Cambridge University Press]] | mr=0116056 | year=1960 | chapter=On the fourteenth problem of Hilbert | pages=459–462}} | |||
*{{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Lectures on the fourteenth problem of Hilbert | url=http://www.math.tifr.res.in/~publ/ln/tifr31.pdf | publisher=Tata Institute of Fundamental Research | location=Bombay | series=Tata Institute of Fundamental Research Lectures on Mathematics | id={{MR|0215828}} | year=1965 | volume=31}} | |||
*{{Citation | last1=Totaro | first1=Burt | title=Hilbert's 14th problem over finite fields and a conjecture on the cone of curves | doi=10.1112/S0010437X08003667 | mr=2457523 | year=2008 | journal=Compositio Mathematica | issn=0010-437X | volume=144 | issue=5 | pages=1176–1198}} | |||
* O. Zariski, ''Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert'', Bulletin des Sciences Mathematiques 78 (1954), pp. 155–168. | |||
<references/> | |||
{{Hilbert's problems}} | |||
[[Category:Hilbert's problems|#14]] | |||
[[Category:Invariant theory]] | |||
Revision as of 00:15, 27 November 2013
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated.
The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables,
- k(x1, ..., xn ) over k.
Consider now the ring R defined as the intersection
![R:=K\cap k[x_{1},\dots ,x_{n}]\ .](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d082eb2576f3e3e6d72d22a7c2832d65b0e028)
Hilbert conjectured that all such subrings are finitely generated. It can be shown that the field K is always finitely generated as a field, in other words, there exist finitely many elements
- yi, i = 1 ,...,d in K
such that every element in R can be rationally represented by the yi. But this does not imply that the ring R is finitely generated as a ring, even if all the elements yi could be chosen from R.
After some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954) then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group.
History
The problem originally arose in algebraic invariant theory. Here the ring R is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group over a field k acting algebraically on a polynomial ring k[x1, ..., xn] (or more generally, on a finitely generated algebra defined over a field). In this situation the field K is the field of rational functions (quotients of polynomials) in the variables xi which are invariant under the given action of the algebraic group, the ring R is the ring of polynomials which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert) of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups (in particular the general linear group over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. This finiteness result was later extended by Hermann Weyl to the class of all semi-simple Lie-groups. A major ingredient in Hilbert's proof is the Hilbert basis theorem applied to the ideal inside the polynomial ring generated by the invariants.
Zariski's formulation
Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k.
Zariski's formulation was shown[1] to be equivalent to the original problem, for X normal.
Nagata's counterexample
Template:Harvtxt gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a1i, ...,a16i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R is the polynomial ring k[x1,...,x16, t1,...,t16] in 32 variables. The vector space V is a 13-dimensional vector space over k consisting of all vectors (b1,...,b16) in k16 orthogonal to each of the three vectors (a1i, ...,a16i) for i=1, 2, 3. The vector space V is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on R by fixing all elements tj and taking xj to xj + bjtj. Then the ring of elements of R invariant under the action of the group V is not a finitely generated k-algebra.
Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example, Template:Harvtxt showed that over any field there is an action of the sum GTemplate:Su of three copies of the additive group on k18 whose ring of invariants is not finitely generated.
References
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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 - 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 - O. Zariski, Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert, Bulletin des Sciences Mathematiques 78 (1954), pp. 155–168.
- ↑ 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