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A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which naturally arises from a polynomial vector field on $\mathbb{R}^{2}$.

According to the comment of Joonas on this question , 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 $S^{2}$?

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