Characterization of weakly regular p-ary bent functions of $$\ell $$ -form

We study the essential properties of weakly regular p-ary bent functions of \(\ell \)-form, where a p-ary function is from \(\mathbb {F}_{p^m}\) to \(\mathbb {F}_p\). We observe that most of studies on a weakly regular p-ary bent function f with \(f(0)=0\) of \(\ell \)-form always assume the gcd-condition: \(\gcd (\ell -1,p-1)=1\). We first show that whenever considering weakly regular p-ary bent functions f with \(f(0) = 0\) of \(\ell \)-form, we can drop the gcd-condition; using the gcd-condition, we also obtain a characterization of a weakly regular bent function of \(\ell \)-form. Furthermore, we find an additional characterization for weakly regular bent functions of \(\ell \)-form; we consider two cases m being even or odd. Let f be a weakly regular bent function of \(\ell \)-form preserving the zero element; then in the case that m is odd, we show that f satisfies \(\gcd (\ell ,p-1)=2\). On the other hand, when m is even and f is also non-regular, we show that f satisfies \(\gcd (\ell ,p-1)=2\) as well. In addition, we present two explicit families of regular bent functions of \(\ell \)-form in terms of the gcd-condition.