  The main contribution of this paper is to find a representation of the class
$\mathcal{F}_d(p)$ of multivariate Bernoulli distributions with the same mean
$p$ that allows us to find its generators analytically in any dimension. We map
$\mathcal{F}_d(p)$ to an ideal of points and we prove that the class
$\mathcal{F}_d(p)$ can be generated from a finite set of simple polynomials. We
present two applications. Firstly, we show that polynomial generators help to
find extremal points of the convex polytope $\mathcal{F}_d(p)$ in high
dimensions. Secondly, we solve the problem of determining the lower bounds in
the convex order for sums of multivariate Bernoulli distributions with given
margins, but with an unspecified dependence structure.
