Group algebra

Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics.

The original problem was posed as the Problem of the topology of algebraic curves and surfaces (Problem der Topologie algebraischer Kurven und Flächen).

Actually the problem consists of two similar problems in different branches of mathematics:

• An investigation of the relative positions of the branches of real algebraic curves of degree n (and similarly for algebraic surfaces).
• The determination of the upper bound for the number of limit cycles in two-dimensionial polynomial vector fields of degree n and an investigation of their relative positions.

The first problem is yet unsolved for n = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. The second problem also remains unsolved: no upper bound for the number of limit cycles is known for any n>1, and this is what usually is meant by Hilbert's sixteenth problem in the field of dynamical systems.

The first part of Hilbert's 16th problem

In 1876 Harnack investigated algebraic curves in the real projective plane and found that curves of degree n could have no more than

${n^{2}-3n+4 \over 2}$

separate connected components. Furthermore he showed how to construct curves that attained that upper bound, and thus that it was the best possible bound. Curves with that number of components are called M-curves.

Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.

Furthermore he requested a generalization of Harnack's Theorem to algebraic surfaces and a similar investigation of surfaces with the maximum number of components.

The second part of Hilbert's 16th problem

Here we are going to consider polynomial vector fields in the real plane, that is a system of differential equations of the form:

${dx \over dt}=P(x,y),\qquad {dy \over dt}=Q(x,y)$

where both P and Q are real polynomials of degree n.

These polynomial vector fields were studied by Poincaré, who had the idea of abandoning the search for finding exact solutions to the system, and instead attempted to study the qualitative features of the collection of all possible solutions.

Among many important discoveries, he found that the limit sets of such solutions need not be a stationary point, but could rather be a periodic solution. Such solutions are called limit cycles.

The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree n and, similar to the first part, investigate their relative positions.

Results

It was shown in 1991/1992 by Yulii Ilyashenko and Jean Écalle that every polynomial vector field in the plane has only finitely many limit cycles (a 1923 article by Henri Dulac claiming a proof of this statement had been shown to contain a gap in 1981). This statement is not obvious, since it is easy to construct smooth (C^∞) vector fields in the plane with infinitely many concentric limit cycles.

The question whether there exists a finite upper bound H(n) for the number of limit cycles of planar polynomial vector fields of degree n remains unsolved for any n>1. (H(1) = 0 since linear vector fields do not have limit cycles.) Evgenii Landis and Ivan Petrovsky claimed a solution in the 1950s, but it was shown wrong in the early 1960s. Quadratic plane vector fields with four limit cycles are known.

The original formulation of the problems

In his speech, Hilbert presented the problems as:

Benefits of Living in a Rental or Landed property in Singapore Property New Launches & Undertaking Showcase In Singapore Many residential Singapore property gross sales contain shopping for property in Singapore at new launches. These are often properties below development, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, in accordance with personal property transactions in Could Cellular Apps FREE Signal Up Log in Property Agents Feedback

Banks usually engage skilled valuers to conduct valuations of properties. The valuers are required to abide by industry requirements and guidelines for property valuation prescribed by the Singapore Institute of Surveyors and Valuers. With Singapore conserving its greenery and with town sky view being the attraction to many, it is no wonder that increasingly condos are being developed on the island. Many future traders are placing extra value on units with a view than these with out one. Thus, this may be one of the reasons too as to why your unit of selection is either with a view or with out. How to Spot Abroad Property Scams (at Propwise.sg) New Condominium Eco Sanctuary Launch 01Dec 2012 Coming Quickly !!! New launch rental COCO Palm at Pasir Ris

Association charges has to be paid by each condominium proprietor, weather you are dwelling in it or not, or even in case you have rented the property. So, whether you employ the amenities and amenities at the condominium, you'll any how must pay for them. JOHOR BAHARU, Jan 22 (Bernama) — The property market in Johor Baharu and Iskandar Malaysia has been negatively affected by the cooling measures introduced by the authorities to address rising property prices, says the Actual Property and Housing Developers' Association of Malaysia (Rehda). District 18, 99 years LH Executive Apartment District 25, ninety nine years LH Executive Rental District 19, ninety nine years LH Government Apartment pipelines. Property homeowners haven't any different but to pay up. There's no different sensible Hottest Launch D'Leedon

Dr Victor Lee is a surgeon gazetted to carry out multi-organ transplant surgical procedure. He underwent his fellowship coaching in transplant surgery on the Royal Infirmary of Edinburgh, United Kingdom in 2008 and 2009. He is presently a consultant surgeon at the Singapore Normal Hospital, and visiting guide to the National College Hospital. His scientific interest and skills are in the areas of transplant surgery, hepatobiliary pancreatic surgery and laparoscopic surgical procedure.

Lush Acres Government Condominium goes to launch quickly at Sengkang West Method By Metropolis Developments Limited. Lush Acres reflects the large expense of house, environmentally-inexperienced constructing and plush panorama of this low density development with solely 22% of website protection. There have been students, retirees and property agents napping, snacking and taking part in cards within the queue to kill time. Such a scene made passers-by wonder if the gang really consisted of genuine consumers, or just part-timers being paid to fake how sizzling the new undertaking was. The model was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France at the INPI (National Institute of Intellectual Property) Coastal commercial spaceports

Individuals all wish to be seen having the identical foresight because the specialists in property investment or the massive names in their own fields. Thus the discharge of these tales works to motivate other patrons to follow go well with. Bartley Ridge is the hottest new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-yr leasehold new condo at Mount Vernon highway, excellent subsequent to Bartley MRT station (CC12). If you wish to get more Rehda Johor chairman Koh Moo Hing stated potential property patrons within the two areas at the moment are adopting a wait-and-see angle. How can I get the ebrochure and flooring plans of the new launch initiatives ? The Current Mortgage to your HDB District thirteen, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

The simplest way is to contact a real estate agent who's a Advertising Workforce Member The agent will doubtless meet to show you details of the undertaking supplied by the developer. Attending a preview doesn't suggest you are obliged to purchase a unit. You can select not to and your cheque will likely be returned to you. No cost concerned. Your cheque will likely be returned to you in the event you decide to not proceed to purchase. Early hen low cost (up to 10%) at the preview earlier than the official launch. The purpose of that is to stimulate demand and to create a buying house in singapore momentum. As soon as this function is achieved, the developer would then take away the low cost so as to maximize his profitability. 2/3/4 bedrooms units obtainable ! GREATEST PURCHASE NOW !!! AVG $1900psf The Key Commercial Cluster in the North Area

Hilbert continues:

Benefits of Living in a Rental or Landed property in Singapore Property New Launches & Undertaking Showcase In Singapore Many residential Singapore property gross sales contain shopping for property in Singapore at new launches. These are often properties below development, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, in accordance with personal property transactions in Could Cellular Apps FREE Signal Up Log in Property Agents Feedback

Banks usually engage skilled valuers to conduct valuations of properties. The valuers are required to abide by industry requirements and guidelines for property valuation prescribed by the Singapore Institute of Surveyors and Valuers. With Singapore conserving its greenery and with town sky view being the attraction to many, it is no wonder that increasingly condos are being developed on the island. Many future traders are placing extra value on units with a view than these with out one. Thus, this may be one of the reasons too as to why your unit of selection is either with a view or with out. How to Spot Abroad Property Scams (at Propwise.sg) New Condominium Eco Sanctuary Launch 01Dec 2012 Coming Quickly !!! New launch rental COCO Palm at Pasir Ris

Association charges has to be paid by each condominium proprietor, weather you are dwelling in it or not, or even in case you have rented the property. So, whether you employ the amenities and amenities at the condominium, you'll any how must pay for them. JOHOR BAHARU, Jan 22 (Bernama) — The property market in Johor Baharu and Iskandar Malaysia has been negatively affected by the cooling measures introduced by the authorities to address rising property prices, says the Actual Property and Housing Developers' Association of Malaysia (Rehda). District 18, 99 years LH Executive Apartment District 25, ninety nine years LH Executive Rental District 19, ninety nine years LH Government Apartment pipelines. Property homeowners haven't any different but to pay up. There's no different sensible Hottest Launch D'Leedon

Dr Victor Lee is a surgeon gazetted to carry out multi-organ transplant surgical procedure. He underwent his fellowship coaching in transplant surgery on the Royal Infirmary of Edinburgh, United Kingdom in 2008 and 2009. He is presently a consultant surgeon at the Singapore Normal Hospital, and visiting guide to the National College Hospital. His scientific interest and skills are in the areas of transplant surgery, hepatobiliary pancreatic surgery and laparoscopic surgical procedure.

Lush Acres Government Condominium goes to launch quickly at Sengkang West Method By Metropolis Developments Limited. Lush Acres reflects the large expense of house, environmentally-inexperienced constructing and plush panorama of this low density development with solely 22% of website protection. There have been students, retirees and property agents napping, snacking and taking part in cards within the queue to kill time. Such a scene made passers-by wonder if the gang really consisted of genuine consumers, or just part-timers being paid to fake how sizzling the new undertaking was. The model was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France at the INPI (National Institute of Intellectual Property) Coastal commercial spaceports

Individuals all wish to be seen having the identical foresight because the specialists in property investment or the massive names in their own fields. Thus the discharge of these tales works to motivate other patrons to follow go well with. Bartley Ridge is the hottest new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-yr leasehold new condo at Mount Vernon highway, excellent subsequent to Bartley MRT station (CC12). If you wish to get more Rehda Johor chairman Koh Moo Hing stated potential property patrons within the two areas at the moment are adopting a wait-and-see angle. How can I get the ebrochure and flooring plans of the new launch initiatives ? The Current Mortgage to your HDB District thirteen, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

The simplest way is to contact a real estate agent who's a Advertising Workforce Member The agent will doubtless meet to show you details of the undertaking supplied by the developer. Attending a preview doesn't suggest you are obliged to purchase a unit. You can select not to and your cheque will likely be returned to you. No cost concerned. Your cheque will likely be returned to you in the event you decide to not proceed to purchase. Early hen low cost (up to 10%) at the preview earlier than the official launch. The purpose of that is to stimulate demand and to create a buying house in singapore momentum. As soon as this function is achieved, the developer would then take away the low cost so as to maximize his profitability. 2/3/4 bedrooms units obtainable ! GREATEST PURCHASE NOW !!! AVG $1900psf The Key Commercial Cluster in the North Area

External links

• 16th Hilbert problem: computation of Lyapunov quantities and limit cycles in two-dimensional dynamical systems