Polynomial solutions of quasi-homogeneous partial differential equations  

作　　者：LUO Xuebo ZHENG Zhujun Institute of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China Institute of Mathematics, Henan University, Kaifeng 475001, China 

出　　处：《Science China Mathematics》2001年第9期1148-1155,共8页中国科学：数学（英文版）

基　　金：the National Natural Science Foundation of China (Grnat No. 19971068) .

摘　　要：By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation $\left\{ {\delta _\tau } \right\}{\text{ }}_{\tau< 0} $ given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or $m{\text{ = }}\sum\limits_{j = 1}^n {\alpha _j \alpha _j } $ for some $\alpha {\text{ = }}\left( {\alpha _1 ,{\text{ }} \ldots {\text{ }},\alpha _n } \right) \in l _ + ^n $ Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ?n must be infiniteBy 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.t a dilation {δ-τ}-{τ<0} given by (a-1,…, a-n). Assume that either a1,…, an are positive rational numbers or m=∑nj=1α-jα-j for some α=(α-1, …, αn)∈I{}+n-+. Then the dimension of the space of polynomial solutions of the equation p[u]=0 on R+n must be infinite.

关 键 词：quasi-homogeneous partial differential operator polynomial solution  dimension of the space of solution method of analytic number theory 

分 类 号：O175[理学—数学]