We introduce a systematic framework for counting and finding independentoperators in effective field theories, taking into account the redundanciesassociated with use of the classical equations of motion and integration byparts. By working in momentum space, we show that the enumeration problem canbe mapped onto that of understanding a polynomial ring in the field momenta.All-order information about the number of independent operators in an effectivefield theory is encoded in a geometrical object of the ring known as theHilbert series. We obtain the Hilbert series for the theory of N real scalarfields in (0+1) dimensions--an example, free of space-time and internalsymmetries, where aspects of our framework are most transparent. Although thisis as simple a theory involving derivatives as one could imagine, it providesfruitful lessons to be carried into studies of more complicated theories: wefind surprising and rich structure from an interplay between integration byparts and equations of motion and a connection with SL(2,C) representationtheory which controls the structure of the operator basis.
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