A multivariate cryptograpic instance in practice is a multivariate polynomial system. So the security of a protocol rely on the complexity of solving a multivariate polynomial system. In this paper there is an overview on a general algorithm used to solve a multivariate system and the quantity to which the complexity of this algorithm depends on: the solving degree. Unfortunately, it is hard to compute. For this reason, it is introduced an invariant: the degree of regularity. This invariant, under certain condition, give us an upper bound on the solving degree. Then we speak about random polynomial systems and in particular what "random" means to us. Finally, we give an upper bound on both the degree of regularity and the solving degree of such random systems.

翻译：在实际应用中，多元密码学实例即为多元多项式系统。因此协议的安全性取决于求解多元多项式系统的复杂性。本文概述了求解多元系统的通用算法及其复杂性所依赖的量——求解次数。遗憾的是，该量难以直接计算。为此，我们引入了一个不变量：正则次数。该不变量在特定条件下给出了求解次数的一个上界。随后讨论了随机多项式系统，特别是"随机"一词在我们语境下的含义。最后给出了此类随机系统的正则次数与求解次数的上界。