I have a multivariate polynomial $P$ which is a product of $M$ low degree polynomials $p_i$

$$P(x_1, x_2, \dotsc, x_n) = \prod_{i=1}^M p_i(x_1, x_2, \dotsc, x_n)$$

where the maximum degree of each $p_i$ can be $4$. For example:

$$P(x_1, x_2, x_3) = (1+x_1x_2^2x_3)(x_3^3+x_1x_2^3).$$

I need to find out whether a specific term $x_1^{m_1}x_2^{m_2}\dotsm x_n^{m_n}$ appears in $P$ or not. Can I find the answer without expanding the products explicitly?