A polynomial vector field  of  degree $n$ on $S^{2}$ is  the  [Poincare compactification](http://www.scielo.cl/pdf/proy/v22n3/art01.pdf) of a $n$ degree polynomial vector  field on $\mathbb{R}^{2}$.

According to the  comment of  Joonas on [this question](http://mathoverflow.net/questions/182415/elliptic-operators-corresponds-to-non-vanishing-vector-fields)
, we  ask:


>Is  there uniform upper bound $IH(n)$, depending only on $n$,  for  the dimension of  kernel and  cokernel of the  elliptic  diff operator $D_{X} +\Delta$ on $S^{2}$  where $X$  is  a  polynomial  vector  field  of  degree $n$  on  $S^{2}$?