On a class of degenerate hypoelliptic polynomials

This work is devoted to finding conditions on “lower-order terms”, the addition of which does not violate the hypoellipticity of a given operator (of the characteristic polynomial of this operator). Necessary and sufficient conditions for the hypoellipticity of “two-layered polynomials” are obtained. In terms of comparing strength, power, and the upper bound of the ratio of comparable polynomials, conditions are obtained under which the added polynomials preserve the hypoellipticity of the original polynomial. Examples are given that illustrate the obtained results.