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SUMMARY:Dmitry Khavinson (University of South Florida)
DTSTART:20210518T130000Z
DTEND:20210518T140000Z
DTSTAMP:20260422T201358Z
UID:CAvid/41
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/41/">A
 lgebra and PDE :  Some Less Traveled Paths Connecting Them</a>\nby Dmitry 
 Khavinson (University of South Florida) as part of CAvid: Complex Analysis
  video seminar\n\nLecture held in N/A.\n\nAbstract\nHere are samples of qu
 estions I plan to discuss.\n\n- Let $F(u\,v)$ be a rational function of tw
 o variables that has no linear factors and a meromorphic function $u(x\,y)
 $ solves the PDE $F(\\nabla u)=0$ near the origin\, say. Then $u$ is a lin
 ear function\, i.e.\, $u=ax+by+c$. Why?  Is it true in three variables?\n\
 n- Does there exist a harmonic polynomial in $\\mathbb{R}^n$ divisible by 
 a non-negative polynomial?\n\n- Let $P(D)[u^k]=0$\, where $P(D)$ is a part
 ial differential operator with constant\, polynomial \, or even entire coe
 fficients and k runs over an arithmetic progression of positive integers\,
  e. g.\, $k=2n+3$\, $n=1\,2\,\\ldots$.  Then the Hessian\, Hess $u$\, vani
 shes identically\, so the mapping grad $u:\\\, \\mathbb{C}^n\\mapsto\\math
 bb{C^n}$ is degenerate\, i.e.\, the range is an algebraic variety. Is it t
 rue? \n\n- When we are solving the Dirichlet problem in a domain with an a
 lgebraic boundary\, and the Dirichlet data is a polynomial\, a rational or
  an algebraic function\, is the solution algebraic as well?\n
LOCATION:https://researchseminars.org/talk/CAvid/41/
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