On strongly hypoelliptic polynomials (Q1382909)

The author studies strongly hypoelliptic polynomials. A multivariate polynomial \(P\) is strongly hypoelliptic if \(P(\xi)\rightarrow\infty\) as \(\| \xi\| \rightarrow\infty\) and there exist constants \(c,M>0\) such that \(| P(\xi)| \leq cd_P^m(\xi)\), \(\xi\in\mathbb R^n\), \(\| \xi\| \geq M\), \(m=\text{ord}(P)\equiv \max_{\alpha\in(P)}| \alpha| \), where \(d_P(\xi)\) is the distance from the point \(\xi\) to the set \({\mathcal D}(P)=\{\theta\in\mathbb C^n\); \(P(\theta)=0\}\). In the main result there are described sufficient and necessary conditions for a polynomial to be strongly hypoelliptic. The conditions are obtained from the study of linear transformations of strongly hypoelliptic polynomials and of the growth and strong hypoellipticity.