  We explore the properties of polynomial Lagrangians for chiral $p$-forms
previously proposed by the last named author, and in particular, provide a
self-contained treatment of the symmetries and equations of motion that shows a
great economy and simplicity of this formalism. We further use analogous
techniques to construct polynomial democratic Lagrangians for general $p$-forms
where electric and magnetic potentials appear on equal footing as explicit
dynamical variables. Due to our reliance on the differential form notation, the
construction is compact and universally valid for forms of all ranks, in any
number of dimensions.
