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

High Energy Physics - Theory;High Energy Physics - Phenomenology