The zero-distribution of the Fourier integral $${int}^{infty}_{-{infty}};Q(u)e^{p(u)+^{izu}du$$, where P is a polynomial with leading term $-u^{2m}(m;{geq};1)$ and Q an arbitrary polynomial, is described. To this end, an asymptotic formula for the integral is established by applying the saddle point method.