We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken, Mittmann and Saxena (2013), and exploited by them, and Agrawal, Saha, Saptharishi and Saxena (2016) to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey, Sinhababu and Saxena (2018), we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. and Agrawal et al. in the positive characteristic setting.

翻译：我们研究了代数等级或超度在有限领域保持同质性的问题,这一概念最初由Beecken、Mittmann和Saxena(2013年)和Agrawal、Saha、Saptharishi和Saxena(2013年)提出,并由他们加以利用,用Jacobian标准设计基于代数的独立身份测试,使用Jacobian标准对零特性的特性领域进行。由于Jacob标准对有限特性领域不起作用,因此无法对有限特性领域进行此类构造的类似性。根据Pandey、Sinhababu和Saxena(2018年)的最新标准,我们在正性特征领域设置中为某些自然类的多元性类别绘制了明确忠实的地图,其中某些参数称为多元性基础的密不可分(这一参数在特性零领域总是为1),这是Beecken等人和Agrawal等人在正性特征环境中的一些结果的首个概括。