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SUMMARY:r(x\,y) + q(x\,y) p(x\,y) = f(x) - g(y)
DTSTART:20250612T140000Z
DTEND:20250612T160000Z
DTSTAMP:20260610T234800Z
UID:indico-event-14371@indico.math.cnrs.fr
DESCRIPTION:Speakers: Manfred Buchacher (Johannes Kepler Universität Linz
 )\n\nI will talk about my research on finding the solutions of such equat
 ions when r(x\,y)\, q(x\,y)\, p(x\,y)\, f(x) and g(y) are rational functi
 ons of which the (irreducible) polynomial p(x\,y) and the rational functio
 n r(x\,y) are known.\n \nAt first glance\, it might not be clear what the
  meaning of the equation is and why one should find it interesting. Howeve
 r\, there is a field theoretic interpretation that is helpful. Let C(x\,y)
  be the field over C that is generated by elements x and y satisfying the 
 relation p(x\,y) = 0\, and let C(x) and C(y) be the subfields generated by
  x and y\, respectively. Then the above equation has a solution if and onl
 y if the element r(x\,y) of C(x\,y) is an element of C(x) + C(y). Note tha
 t when r = 0\, the problem is the problem of computing the intersection of
  C(x) and C(y)\, and that when g = 0\, finding a solution of the equation 
 answers whether r(x\,y) is an element of C(x) and how it can be expressed 
 in terms of x.\n \nThe problem originally arose in enumerative combinator
 ics\, in the enumeration of restricted lattice walks. Other applications a
 re parameter identification problems in ODE models\, or problems of image 
 recognition\, for instance. There are most likely many more as it is a bas
 ic problem in computational algebra and algebraic geometry. \n \nI will 
 present a semi-algorithm that solves such equations. "Semi" means that it 
 might not terminate. It terminates if the equation has a non-trivial solut
 ion. However\, if there is no such solution\, it might not. Its terminatio
 n depends on a dynamical system related to the curve associated with p(x\,
 y) and the location of the poles of r(x\,y).\n \nThe methods and argument
 s employed are mostly elementary\, and come from linear algebra\, the Newt
 on-Puiseux algorithm\, and Galois theory\, and can be interpreted as a com
 bination of tropical (in the sense of tropical geometry) and Galois theore
 tic arguments. The question how the semi-algorithm could be turned into an
  algorithm connects it to number theory and problems in arithmetic / algeb
 raic dynamics.\n \nThere will be pictures.\n\nhttps://indico.math.cnrs.fr
 /event/14371/
LOCATION:207 (Bat 1R2)
URL:https://indico.math.cnrs.fr/event/14371/
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