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.

我们研究\(\ell \)形式的弱正则p元弯曲函数的基本性质，其中p元函数是从\(\mathbb {F}_{p^m}\)到\(\ mathbb {F}_p\)。我们观察到大多数关于具有\(\ell \)形式的\(f(0)=0\)的弱正则p -ary 弯曲函数f的研究总是假设gcd 条件：\(\gcd (\ell -1,p-1)=1\)。我们首先表明，每当考虑具有\(\ell \)形式的\(f(0) = 0\)的弱正则p进制弯曲函数f时，我们可以删除 gcd 条件；使用 gcd 条件，我们还获得了\(\ell \)形式的弱正则弯曲函数的表征。此外，我们发现了\(\ell \)形式的弱正则弯曲函数的附加表征；我们考虑两种情况m是偶数或奇数。令f为保留零元素的\(\ell \)形式的弱正则弯曲函数；那么在m为奇数的情况下，我们证明f满足\(\gcd (\ell ,p-1)=2\)。另一方面，当m是偶数且f也是非正则时，我们证明f也满足\(\gcd (\ell ,p-1)=2\) 。此外，我们根据 gcd 条件提出了两个显式的\(\ell \)形式的正则弯曲函数族。