Constructing permutation polynomials over finite fields, particularly those with simple algebraic structure in multiple variables, is a fundamental problem with applications in cryptography and coding theory. Recently, Li and Kaleyski (IEEE Trans. Inf. Theory, 2024) generalized two sporadic quadratic APN permutations into infinite families of trivariate functions. Motivated by their work, we investigate conditions under which generalized trivariate functions fail to be permutations. We establish necessary conditions on coefficient parameters that prevent the permutation property, provide a complete computational classification for small field extensions, and prove general non-permutation results. As a key application of our algebraic geometry approach, we resolve the permutation part of a conjecture by Beierle, Carlet, Leander, and Perrin (Finite Fields Appl., 2022) regarding a related trivariate form. Specifically, we prove that for all odd characteristic-2 extension degrees $m \geq 23$, their function $C_u$ is not a permutation over $\mathbb{F}_{2^m}^3$ for any $u \in \mathbb{F}_{2^m}^*$, resolving the permutation part of their conjecture for sufficiently large fields.

翻译：在有限域上构造置换多项式，特别是那些在多个变量中具有简单代数结构的置换多项式，是一个基础性问题，在密码学和编码理论中具有重要应用。最近，Li和Kaleyski（IEEE Trans. Inf. Theory, 2024）将两个零散的二次APN置换推广为无限族的三元函数。受其工作启发，我们研究了广义三元函数不构成置换的条件。我们建立了阻止置换性质的系数参数必要条件，为小域扩展提供了完整的计算分类，并证明了广义的非置换结果。作为我们代数几何方法的一个关键应用，我们解决了Beierle、Carlet、Leander和Perrin（Finite Fields Appl., 2022）关于一个相关三元形式猜想中的置换部分。具体而言，我们证明对于所有奇特征2扩展次数$m \geq 23$，他们的函数$C_u$对于任何$u \in \mathbb{F}_{2^m}^*$都不是$\mathbb{F}_{2^m}^3$上的置换，从而在足够大的域上解决了他们猜想中的置换部分。