  In this paper, it is shown that every univariate polynomial function is mixed
monotone globally with a polynomial decomposition function. The decomposition
functions can be constructed from the Gram matrix representation of polynomial
functions. The tightness of polynomial decomposition functions is discussed. An
example is provided to show that polynomial decomposition functions, in
addition to being global decomposition functions, can be much tighter than
local decomposition functions constructed using local Jacobian bounds.
Furthermore, an example is provided to demonstrate the application to reachable
set over-approximation.
