Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract    introduce the notion of   m  -sparse por series (e.g. expanding   sin    x   and   cos    x   at   x=0   gives 2-sparse por series: a coefficient   a     n    of the series can be nonzero only if  remainder    (n,2)   is equal to a fixed number). Then  consider the problem of finding all   m  -points of a linear ordinary differential equation   Ly=0   with polynomial coefficients (i.e., the points at which the equation has a solution in the form of an   m  -sparse series). It is easy to find an upper bound for   m  .  prove that if   m   is fixed then either there exists a finite number of   m  -points and all of them can be found or all points are   m  -points and   L   can be factored as   L=  L  ̃  ∘C   where   C   is an operator of a special kind with constant coefficients. Additionally,  formulate simple necessary and sufficient conditions for the existence of   m  -points for an irreducible   L  .
['Sergei A. Abramov']
I am writing a research paper on a idea m -sparse solutions of linear ordinary differential equations with polynomial coefficients
[{"from": "human", "value": "I am writing a research paper on a idea m -sparse solutions of linear ordinary differential equations with polynomial coefficients"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract    introduce the notion of   m  -sparse por series (e.g. expanding   sin    x   and   cos    x   at   x=0   gives 2-sparse por series: a coefficient   a     n    of the series can be nonzero only if  remainder    (n,2)   is equal to a fixed number). Then  consider the problem of finding all   m  -points of a linear ordinary differential equation   Ly=0   with polynomial coefficients (i.e., the points at which the equation has a solution in the form of an   m  -sparse series). It is easy to find an upper bound for   m  .  prove that if   m   is fixed then either there exists a finite number of   m  -points and all of them can be found or all points are   m  -points and   L   can be factored as   L=  L  \u0303  \u2218C   where   C   is an operator of a special kind with constant coefficients. Additionally,  formulate simple necessary and sufficient conditions for the existence of   m  -points for an irreducible   L  ."}]