On the (non-)existence of APN (n, n)-functions of algebraic degree n

We study the problem of existence of APN functions of algebraic degree n over F
_2
n
. We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce some non-existence results which mean, in particular, that for most of the known APN functions F over F
_2
n
the function x
^2
n
-1
+ F(x) is not APN, and changing a value of F in a single point results in non-APN functions.