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 ofcharacteristic 0 or 2. These descriptions are given in the form of decomposition schemes of polynomial functions. Similar descriptions are givenfor arbitrary finite fields. However, we show that these descriptions do not extend to infinite fields of odd characteristic.