Polynomial solutions of quasi-homogeneous partial differential equations

<正>By means of a method of analytic number theory the following theorem is proved. Let p be a quasi-homogeneous linear partial differential operator with degree m, m > 0, w.r.ta dilation{δτ}τ<0 given by ( a1, …, an). Assume that either a1, …,an are positive rational numbers or m =ajaj for some a = (a1,…,an)∈In+. Then the dimension of the space of polynomial solutions of the equation p[u] =0 on Rn must be infinite.