Some results on the vanishing conjecture of differential operators with constant coefficients

In this paper we prove four cases of the Vanishing Conjecture of differential operators with constant coefficients and also a conjecture on the Laurent polynomials with no holomorphic parts, which were proposed in [15] by the third named author. We also give two examples to show that both the Vanishing Conjecture and the Duistermaat–van der Kallen Theorem [6] cannot be generalized to the setting of (Laurent) formal power series in general.