Algebra and PDE : Some Less Traveled Paths Connecting Them

Abstract: Here are samples of questions I plan to discuss.

- Let $F(u,v)$ be a rational function of two 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 linear function, i.e., $u=ax+by+c$. Why? Is it true in three variables?

- Does there exist a harmonic polynomial in $\mathbb{R}^n$ divisible by a non-negative polynomial?

- Let $P(D)[u^k]=0$, where $P(D)$ is a partial differential operator with constant, polynomial , or even entire coefficients and k runs over an arithmetic progression of positive integers, e. g., $k=2n+3$, $n=1,2,\ldots$. Then the Hessian, Hess $u$, vanishes identically, so the mapping grad $u:\, \mathbb{C}^n\mapsto\mathbb{C^n}$ is degenerate, i.e., the range is an algebraic variety. Is it true?

- When we are solving the Dirichlet problem in a domain with an algebraic boundary, and the Dirichlet data is a polynomial, a rational or an algebraic function, is the solution algebraic as well?

analysis of PDEscomplex variables

Audience: researchers in the topic

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CAvid: Complex Analysis video seminar

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