On the Gabor frame set of polynomial-Gaussian windows

We prove two sufficient conditions for the frame property of Gabor systems $\mathcal{G}(g,a,b)=\{e^{2πi mb\,\cdot}g(\cdot-ak):k,m\in\mathbb{Z} \}$ generated by polynomial–Gaussian windows $g(t)=P(t)e^{-ct^2}$, where $c>0$ and $P$ is a nonzero complex polynomial of degree $n$. The conditions depend only on the product $ab$ of the lattice parameters and the degree of $P$. First, $\mathcal{G}(g,a,b)$ is a frame whenever $ab<1$ is irrational. Second, if $ab=p/q<1$, with $p,q\in\mathbb{N}$ coprime, then $\mathcal{G}(g,a,b)$ is a frame whenever $q-p>n$. Consequently, in the region $ab<1$, failure of the frame property can occur only on rational hyperbolas whose reduced denominator and numerator differ by at most $n$. The proof extends the translated-Wronskian method of Faulhuber and Petersen [The frame set of the first Hermite function, arXiv:2609.02610] from the first Hermite function to arbitrary polynomial multiples of a Gaussian, combining a multiplicity estimate for Wronskians with Jensen's formula.