Abstract : Given real valued polynomials $P$ on $\mathbb{R}^n$ and various unbounded domains $D \subset \mathbb{R}^n$, we consider the oscillatory integrals $$I(P, D, \lambda) = \int_D e^{i\lambda P(t)} dt.$$ We establish a criterion on $(P, D)$ to determine the convergence of these integrals, and find the oscillation indices when they converge. These indices are described in terms of a generalized notion of Newton polyhedra associated with $(P, D)$. When $(P, D)$ for $D=\mathbb{R}^n$ satisfies the criterion of the vector polynomial version $(t_1, \cdots, t_n, P(t))$, we obtain the Strichartz estimates for the following general linear propagators: $e^{it P(D)}(f)(x) \text{ where } D=\left(\frac{\partial_{x_1}}{i}, \cdots, \frac{\partial_{x_n}}{i} \right).$