多变量多项式在有限域上绝对不可约性的新判据
A new criterion for the absolute irreducibility of multivariate polynomials over finite fields
AI总结：
本文提出了一种新的判据，用于判断有限域上由多变量多项式定义的超曲面的绝对不可约性，无需测试基域或扩展域的不可约性，仅需多变量GCD计算和平方自由性质。
摘要：
代数簇的一个关键性质是其是否绝对不可约，即在定义域的代数闭包上仍保持不可约性，确定绝对不可约性在代数几何及其在编码理论、密码学和其他领域中具有重要意义。绝对不可约性的应用包括通过Weil猜想界定点有理点的数量，以及建立有限域上函数的例外APN和置换性质。在本文中，我们提出了一种新的判据，用于判断由多变量多项式定义的超曲面在有限域上的绝对不可约性。我们的判据不需要在基域或扩展域上测试不可约性，假设主项是平方自由的。我们只需要进行多变量GCD计算和平方自由性质的检验。由于几乎所有的多项式已知都是平方自由的，因此我们的绝对不可约性判据适用于几乎所有多变量多项式。
英文摘要：
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.
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