This paper proposes a method to use polynomial forms to represent the distribution of state variables in discrete-time stochastic dynamical systems and provide guaranteed bounds on the distributions of the state variables. The main idea is to combine higher order interval arithmetic and concentration of measure inequalities to quantify uncertainty. This work builds upon the authors previous work that uses affine forms rather than polynomial forms to quantify uncertainty. All reviewers found the paper clearly written and are positively predisposed towards acceptance. One reviewer raised two points relating to comparison to related methods and raised their score after the authors addressed these points in their response. Another reviewer noted post response that while the paper has limitations but it also has merits. As long as the camera ready adequately discusses these limitations, this paper may inspire future work.