Possibility of a function which is not a polynomial, but its laplacian (dimension >=2) is a polynomial

Question

I was going through some literature regarding virtual element function spaces.

I was wondering if it is possible for a function $v_h$ to exist such that $v_h \notin P_k$ but $\Delta v_h \in P_{k-2}$ for dimensions greater than or equal to $2$.

Thanks in advance!

Answer 1

If $v$ is a polynomial, add a harmonic function that is not a polynomial. E.g. in dimension $2$, the real part of a non-polynomial entire function of $z = x+iy$ will do.