Much work has been devoted to bent functions in odd characteristic, but there still remains a gap between our knowledge of binary and nonbinary bent functions. In the first part of this paper, we attempt to partially bridge this gap by generalizing to any characteristic important properties known in characteristic two concerning the Walsh transform of derivatives of bent functions. Some of these properties generalize to all bent functions, while others appear to apply only to weakly regular bent functions. We deduce a method to obtain a bent function by adding a quadratic function to a weakly regular bent function. We also identify a particular class of bent functions possessing the property that every first-order derivative in a nonzero direction has a derivative (which is then a second-order derivative of the function) equal to a nonzero constant. We show that this property implies bentness and is shared in particular by all cubic bent functions. This generalizes to the odd characteristic the notion of cubic-like bent function, that was introduced and studied for binary functions by Irene Villa and the first author. In the second part of the paper, we provide (for the first time) a primary construction leading to an infinite class of cubic ternary vectorial bent functions that have only not weakly regular components. We show the bentness of the component functions by two approaches: by calculating the Walsh transform directly and by considering the second-order derivatives (and applying the results from the first part of the paper). We prove that they are not weakly regular by showing they do not have one of the properties that we proved in the first part of the paper for weakly regular bent functions.

翻译：关于奇特征弯曲函数的研究已有大量工作，但我们对二元与非二元弯曲函数的认知仍存在差距。本文第一部分尝试通过将特征二中关于弯曲函数导数Walsh变换的重要性质推广至任意特征，部分弥合这一差距。其中某些性质适用于所有弯曲函数，另一些则可能仅适用于弱正则弯曲函数。我们由此推导出一种方法：将二次函数与弱正则弯曲函数相加即可得到弯曲函数。我们还识别了一类特殊的弯曲函数，该类函数具有如下性质：任意非零方向的一阶导数的导数（即函数的二阶导数）等于非零常数。我们证明了该性质蕴含弯曲性，并特别为所有三次弯曲函数所共有。这将其推广至奇特征下的三次类弯曲函数概念——该概念最初由Irene Villa与本文第一作者针对二元函数提出并研究。本文第二部分首次给出基本构造，导出一类无穷族的三次三元向量弯曲函数，其所有分量均非弱正则。我们通过两种方法证明分量函数的弯曲性：直接计算Walsh变换以及考虑二阶导数（并应用第一部分的结果）。通过证明其不满足第一部分为弱正则弯曲函数建立的性质之一，我们证实它们不是弱正则的。