On the degree of homogeneous bent functions

It is well known that the degree of a 2m-variable bent function is at most m. However, the case in homogeneous bent functions is not clear. In this paper, it is proved that there is no homogeneous bent functions of degree m in 2m variables when m > 3; there is no homogenous bent function of degree m− 1 in 2m variables when m > 4; Generally, for any nonnegative integer k, there exists a positive integer N such that when m > N , there is no homogeneous bent functions of degree m−k in 2m variables. In other words, we get a tighter upper bound on the degree of homogeneous bent functions. A conjecture is proposed that for any positive integer k > 1, there exists a positive integer N such that when m > N , there exists homogeneous bent function of degree k in 2m variables.