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The '''Ax–Kochen theorem''', named for [[James Ax]] and [[Simon B. Kochen]], states that for each positive integer ''d'' there is a finite set ''Y<sub>d</sub>'' of prime numbers, such that if ''p'' is any prime not in ''Y<sub>d</sub>'' then every homogeneous polynomial of degree ''d'' over the [[p-adic number]]s in at least ''d''<sup>2</sup>+1 variables has a nontrivial zero.<ref>James Ax and Simon Kochen, ''Diophantine problems over local fields I.'', American Journal of Mathematics, '''87''', pages 605-630, (1965)</ref> | |||
==The proof of the theorem== | |||
The proof of the theorem makes extensive use of methods from [[mathematical logic]], such as [[model theory]]. | |||
One first proves [[Serge Lang]]'s theorem, stating that the analogous theorem is true for the field '''F'''<sub>''p''</sub>((''t'')) of formal [[Laurent series]] over a [[finite field]] '''F'''<sub>''p''</sub> with <math>Y_d = \varnothing</math>. In other words, every homogeneous polynomial of degree ''d'' with more than ''d''<sup>2</sup> variables has a non-trivial zero (so '''F'''<sub>''p''</sub>((''t'')) is a [[quasi-algebraically closed field|C<sub>2</sub> field]]). | |||
Then one shows that if two [[Hensel's lemma|Henselian]] [[Valuation (algebra)|valued]] fields have equivalent valuation groups and residue fields, and the residue fields have [[characteristic (algebra)|characteristic]] 0, then they are elementarily equivalent (which means that a first order sentence is true for one if and only if it is true for the other). | |||
Next one applies this to two fields, one given by an [[ultraproduct]] over all primes of the fields '''F'''<sub>''p''</sub>((''t'')) and the other given by an ultraproduct over all primes of the ''p''-adic fields ''Q''<sub>''p''</sub>. | |||
Both residue fields are given by an ultraproduct over the fields '''F'''<sub>''p''</sub>, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are elementarily equivalent. (Taking ultraproducts is used to force the residue field to have characteristic 0; the residue fields of '''F'''<sub>''p''</sub>((''t'')) | |||
and ''Q''<sub>''p''</sub> both have non-zero characteristic ''p''.) | |||
The elementary equivalence of these ultraproducts implies that for any sentence in the language of valued fields, there is a finite set ''Y'' of exceptional primes, such that for any ''p'' not in this set the sentence is true for '''F'''<sub>''p''</sub>((''t'')) if and only if it is true for the field of ''p''-adic numbers. Applying this to the sentence stating that every non-constant homogeneous polynomial of degree ''d'' in at least ''d''<sup>2</sup>+1 variables represents 0, and using Lang's theorem, one gets the Ax–Kochen theorem. | |||
==Alternative proof== | |||
In 2008, [[Jan Denef]] found a purely geometric proof for a conjecture of [[Jean-Louis Colliot-Thélène]] which generalizes the Ax–Kochen theorem. He presented his proof at the "Variétés rationnelles" seminar <ref>http://www.dma.ens.fr/~gille/sem/sem_variete_07-08.html</ref> at École Normale Supérieure in Paris, but the proof<ref>Jan Denef, [http://perswww.kuleuven.be/~u0009256/Denef-Mathematics/denef_papers/ColliotTheleneConjecture.pdf Proof of a conjecture of Colliot-Thélène]</ref> has not been published yet. | |||
==Exceptional primes== | |||
[[Emil Artin]] conjectured this theorem with the finite exceptional set ''Y<sub>d</sub>'' being empty (that is, that all ''p''-adic fields are [[C2 field|C<sub>2</sub>]]), but [[Guy Terjanian]]<ref>{{cite journal | first=Guy | last=Terjanian | authorlink=Guy Terjanian | title=Un contre-example à une conjecture d'Artin | journal=C. R. Acad. Sci. Paris Sér. A-B | volume=262 | page=A612 | year=1966 | zbl=0133.29705 | language=French }}</ref> found the following 2-adic counterexample for ''d'' = 4. Define | |||
:''G''('''x''') = ''G''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) =Σ ''x''<sub>i</sub><sup>4</sup> − Σ<sub>''i''<''j''</sub> ''x''<sub>i</sub><sup>2</sup>''x''<sub>j</sub><sup>2</sup> − ''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>(''x''<sub>1</sub> + ''x''<sub>2</sub> + ''x''<sub>3</sub>). | |||
Then ''G'' has the property that it is 1 mod 4 if some ''x'' is odd, and 0 mod 16 otherwise. It follows easily from this that the homogeneous form | |||
:''G''('''x''') + ''G''('''y''') + ''G''('''z''') + 4''G''('''u''') + 4''G''('''v''') + 4''G''('''w''') | |||
of degree ''d''=4 in 18> ''d''<sup>2</sup> variables has no non-trivial zeros over the 2-adic integers. | |||
Later Terjanian<ref>Guy Terjanian, ''Formes ''p''-adiques anisotropes.'' (French) Journal für die Reine und Angewandte Mathematik, '''313''' (1980), pages 217-220</ref> showed that for each prime ''p'' and multiple ''d''>2 of ''p''(''p''−1), there is a form over the ''p''-adic numbers of degree ''d'' with more than ''d''<sup>2</sup> variables but no nontrivial zeros. In other words, for all ''d''> 2, ''Y<sub>d</sub>'' contains all primes ''p'' such that ''p''(''p''−1) divides ''d''. | |||
{{harvtxt|Brown|1978}} gave an explicit but very large bound for the exceptional set of primes ''p''. If the degree ''d'' is 1, 2, or 3 the exceptional set is empty. {{harvtxt|Heath-Brown|2010}} showed that if ''d''=5 the exceptional set is bounded by 13, and {{harvtxt|Wooley|2008}} showed that for ''d''=7 the exceptional set is bounded by 883 and for ''d''=11 it is bounded by 8053. | |||
==See also== | |||
*[[Artin's conjecture]] | |||
*[[Brauer's theorem on forms]] | |||
*[[quasi-algebraic closure]] | |||
==Notes== | |||
<references/> | |||
==References== | |||
*{{Citation | last1=Brown | first1=Scott Shorey | title=Bounds on transfer principles for algebraically closed and complete discretely valued fields | url=http://books.google.com/books?id=MesICi8orQkC | isbn=978-0-8218-2204-3 | mr=494980 | year=1978 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=15 | issue=204}} | |||
* {{ cite book | last=Chang | first=C.C. | author2-link=Howard Jerome Keisler|last2=Keisler|first2= H. Jerome | publisher=[[Elsevier]] | title=Model Theory | year=1989 | edition=third | isbn=0-7204-0692-7 }} (Corollary 5.4.19) | |||
*{{Citation | last1=Heath-Brown | first1=D. R. | title=Zeros of p-adic forms | doi=10.1112/plms/pdp043 | mr=2595750 | year=2010 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=100 | issue=2 | pages=560–584}} | |||
*{{Citation | last1=Wooley | first1=Trevor D. | title=Artin's conjecture for septic and unidecic forms | doi=10.4064/aa133-1-2 | mr=2413363 | year=2008 | journal=[[Acta Arithmetica]] | issn=0065-1036 | volume=133 | issue=1 | pages=25–35}} | |||
{{DEFAULTSORT:Ax-Kochen theorem}} | |||
[[Category:Number theory]] | |||
[[Category:Model theory]] | |||
[[Category:Theorems in number theory]] | |||
Revision as of 17:55, 20 April 2013
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2+1 variables has a nontrivial zero.[1]
The proof of the theorem
The proof of the theorem makes extensive use of methods from mathematical logic, such as model theory.
One first proves Serge Lang's theorem, stating that the analogous theorem is true for the field Fp((t)) of formal Laurent series over a finite field Fp with . In other words, every homogeneous polynomial of degree d with more than d2 variables has a non-trivial zero (so Fp((t)) is a C2 field).
Then one shows that if two Henselian valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic 0, then they are elementarily equivalent (which means that a first order sentence is true for one if and only if it is true for the other).
Next one applies this to two fields, one given by an ultraproduct over all primes of the fields Fp((t)) and the other given by an ultraproduct over all primes of the p-adic fields Qp. Both residue fields are given by an ultraproduct over the fields Fp, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are elementarily equivalent. (Taking ultraproducts is used to force the residue field to have characteristic 0; the residue fields of Fp((t)) and Qp both have non-zero characteristic p.)
The elementary equivalence of these ultraproducts implies that for any sentence in the language of valued fields, there is a finite set Y of exceptional primes, such that for any p not in this set the sentence is true for Fp((t)) if and only if it is true for the field of p-adic numbers. Applying this to the sentence stating that every non-constant homogeneous polynomial of degree d in at least d2+1 variables represents 0, and using Lang's theorem, one gets the Ax–Kochen theorem.
Alternative proof
In 2008, Jan Denef found a purely geometric proof for a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem. He presented his proof at the "Variétés rationnelles" seminar [2] at École Normale Supérieure in Paris, but the proof[3] has not been published yet.
Exceptional primes
Emil Artin conjectured this theorem with the finite exceptional set Yd being empty (that is, that all p-adic fields are C2), but Guy Terjanian[4] found the following 2-adic counterexample for d = 4. Define
- G(x) = G(x1, x2, x3) =Σ xi4 − Σi<j xi2xj2 − x1x2x3(x1 + x2 + x3).
Then G has the property that it is 1 mod 4 if some x is odd, and 0 mod 16 otherwise. It follows easily from this that the homogeneous form
- G(x) + G(y) + G(z) + 4G(u) + 4G(v) + 4G(w)
of degree d=4 in 18> d2 variables has no non-trivial zeros over the 2-adic integers.
Later Terjanian[5] showed that for each prime p and multiple d>2 of p(p−1), there is a form over the p-adic numbers of degree d with more than d2 variables but no nontrivial zeros. In other words, for all d> 2, Yd contains all primes p such that p(p−1) divides d.
Template:Harvtxt gave an explicit but very large bound for the exceptional set of primes p. If the degree d is 1, 2, or 3 the exceptional set is empty. Template:Harvtxt showed that if d=5 the exceptional set is bounded by 13, and Template:Harvtxt showed that for d=7 the exceptional set is bounded by 883 and for d=11 it is bounded by 8053.
See also
Notes
- ↑ James Ax and Simon Kochen, Diophantine problems over local fields I., American Journal of Mathematics, 87, pages 605-630, (1965)
- ↑ http://www.dma.ens.fr/~gille/sem/sem_variete_07-08.html
- ↑ Jan Denef, Proof of a conjecture of Colliot-Thélène
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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
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