  The authors' previous results on the arity gap of functions of several
variables are refined by considering polynomial functions over arbitrary
fields. We explicitly describe the polynomial functions with arity gap at least
3, as well as the polynomial functions with arity gap equal to 2 for fields of
characteristic 0 or 2. These descriptions are given in the form of
decomposition schemes of polynomial functions. Similar descriptions are given
for arbitrary finite fields. However, we show that these descriptions do not
extend to infinite fields of odd characteristic.
