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 $S^{2}$?