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Seminar on Complex Analysis (Gonchar Seminar)
October 17, 2022 17:00–18:00, Moscow, Steklov Institute, room 110
 


Number of connected components in the space of Pell–Abel equations admitting fixed degree primitive solution

A. B. Bogatyreva, Q. Gendronb

a Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow
b Instituto de Matemáticas UNAM Unidad Oaxaca

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Abstract: Pell–Abel equation is the functional reincarnation of the known diophantine equation
$$ P^2-DQ^2=1, $$
where $P,Q$ and $D$ are complex polynomials. Monic $D$ is known and has no multiple roots; $P$ and $Q$ have to be found. Given $D$, the set of nontrivial solutions $(P,Q)\neq (1,0)$ is generated by the so called primitive solution with minimal $\operatorname{deg}P>0$. We use pictorial calculus of weighted planar graphs to calculate the number of connected components in the space of equations with fixed degrees of $D$ and the primitive solution.

Website: https://mi-ras-ru.zoom.us/j/6119310351?pwd=anpleGlnYVFXNEJnemRYZk5kMWNiQT09

* ID: 611 931 0351. Password: 5MAVBP
 
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