A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry and its applications in coding theory, cryptography, and other fields. Among the applications of absolute irreducibility are bounding the number of rational points via the Weil conjectures and establishing exceptional APN and permutation properties of functions over finite fields. In this article, we present a new criterion for the absolute irreducibility of hypersurfaces defined by multivariate polynomials over finite fields. Our criterion does not require testing for irreducibility in the ground or extension fields, assuming that the leading form is square-free. We just require multivariate GCD computations and the square-free property. Since almost all polynomials are known to be square-free, our absolute irreducibility criterion is valid for almost all multivariate polynomials.

翻译：代数簇的一个关键性质在于其是否绝对不可约，即在其定义域的代数闭包上是否保持不可约性。判定绝对不可约性在代数几何及其在编码理论、密码学等领域的应用中具有重要意义。绝对不可约性的应用包括通过韦伊猜想界定有理点的数量，以及确立有限域上函数的例外APN与置换性质。本文针对有限域上多元多项式定义的超曲面，提出了一种新的绝对不可约性判定准则。该准则不要求在本原域或扩域中进行不可约性检验，仅需假设其首项形式为无平方因子式。我们仅需进行多元最大公因式计算并验证无平方因子性质。由于已知几乎所有多项式均为无平方因子式，本绝对不可约性判定准则对几乎所有多元多项式均适用。