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.
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