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SUMMARY:An ansatz concerning discrete Darboux polynomials
DTSTART;VALUE=DATE-TIME:20230622T150000Z
DTEND;VALUE=DATE-TIME:20230622T153000Z
DTSTAMP;VALUE=DATE-TIME:20260426T214158Z
UID:indico-contribution-47@cern.ch
DESCRIPTION:Speakers: Prof. ROBERTS\, John (School of Mathematics and Stat
 istics\, UNSW Sydney Australia)\nCelledoni et al [1\,2] have illustrated t
 he power of using so-called Darboux polynomials to search for preserved in
 tegrals and measures of a rational map L (including a map arising as a Kah
 an-Hirota-Kimura discretisation of a (Hamiltonian) ODE).  For instance\,  
 integrals are obtained as the ratio of two Darboux polynomials where each 
 Darboux polynomial satisfies an equation of the form P(x') = C(x) P(x)\, w
 here C(x) is a rational function called a cofactor (and prime denotes the 
 image of x under L).  \n\nIn [1\,2]\, C(x) is assumed to take a form invol
 ving factors from the numerator and denominator of the Jacobian determinan
 t of L.  In this talk\,  we confirm this "Jacobian factor ansatz" for C(x)
  using concepts from algebraic geometry and explore its further consequenc
 es.\n\n\n[1] E.Celledoni\, C.Evripidou\, D.I.McLaren\, B.Owren\, G.R.W. Qu
 ispel and B.K.Tapley\, Detecting and determining preserved measures and in
 tegrals of birational maps\,  J. Comput. Dyn. 9 (2022)\, no. 4\, 553–57\
 n\n[2] E.Celledoni\, C.Evripidou\, D.I.McLaren\, B.Owren\, G.R.W. Quispel\
 , B.K.Tapley and P.H. van der Kamp\, Using discrete Darboux polynomials to
  detect and determine preserved measures and integrals of rational maps\, 
  J. Phys. A 52 (2019)\, 11pp.\n\nhttp://indico.fuw.edu.pl/contributionDisp
 lay.py?contribId=47&sessionId=10&confId=67
LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06
URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=47&sessionId
 =10&confId=67
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