Solvable Pell-Abel equations

• 主讲人：Andrei Bogatytrev (INM RAS, MSU, MCFAM )
• 举办方： Beijing-Moscow Mathematics Colloquium
• 时间： 2024-03-29 17:00 - 2024-03-29 18:00
• 地点： online

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Abstract: The Pell–Abel (PA) functional equation  $P^2(x)-D(x)Q^2(x)=1$ is the reincarnation of the famous Diophantine equation in the world of polynomials, which was considered by N.H. Abel in 1826.
The equation arises in various math environments: reduction of Abelian integrals, elliptic billiards, the spectral problem for infinite Jacobi matrices, approximation theory, etc. If the PA equation has a nontrivial solution, then there are infinitely many of them, and all of them are expressed via a primitive
solution $P(x)$ which has a minimal complexity. Using graphical techniques, we find the number of connected components in the space of PA equations with the coefficient  $D(x)$ of a given
degree and having a primitive solution $P(x)$  of another given degree. Some related problems will be  discussed also.  Joint work  with Quentin Gendron (UNAM Institute of Mathematics)

Bio: Professor Andrei Bogatyrev is a leading scientific researcher at the Marchuk Institute for Numerical Mathematics of Russian Academy of Sciences in Moscow and he is also a Professor at the Lomonosov Moscow State Universuty. He graudated from the Moscow Institute of Physics and Technology in 1994 and obtained PhD in in numerical mathematics in 1996. He obtained the Doctor of Physics-Mathematics Sciences (Russian analogue of habilitation) in 2003. In 2009 he was awarded the  Kovalevskaya Prize from the Russian Academy of Sciences. In 2016 he was elected as the Professor of the Russian Academy of Sciences (honorary title). His research interests are in complex analysis and geometry (including Riemann surfaces and moduli),  mathematical physics, functional analysis and function theory, ordinary differential equations, numerical analysis, approximation theory and optimization.