On the Gabor frame set of polynomial-Gaussian windows

We prove two sufficient conditions for the frame property of Gabor systems G(g,a,b)=2πi mb\,·g(·-ak):k,m \ generated by polynomial--Gaussian windows g(t)=P(t)e-ct², 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, G(g,a,b) is a frame whenever $ab<1$ is irrational. Second, if $ab=p/q<1$, with p,q coprime, then 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.