The Khavinson-Shapiro conjecture and polynomial decompositions

作者：Lundberg Erik^*; Render Hermann

来源：Journal of Mathematical Analysis and Applications, 2011, 376(2): 506-513.

DOI：10.1016/j.jmaa.2010.09.069

• Harmonic and polyharmonic polynomials
• Fischer decompositions
• Harmonic divisors
• Algebraic Dirichlet problems

The main result of the paper states the following: Let psi be a polynomial in n variables. Suppose that there exists a constant C > 0 such that any polynomial f has a polynomial decomposition f = psi q(f) + h(f) with Delta(k)h(f) = 0 and deg q(f) <= deg f + C. Then deg psi <= 2k. Here Delta(k) is the kth iterate of the Laplace operator Delta. As an application, new classes of domains in R-n are identified for which the Khavinson-Shapiro conjecture holds.

• 出版日期2011-4-15