Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptography. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residual field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gr\"obner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Pl\"ucker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

翻译：有限环上的代数几何与编码理论中的若干问题可建模为代数方程组。其中包含用于构造公钥密码的秩译码问题。2004年，Nechaev与Mikhailov提出了求解有限链环上多项式方程组的两种方法，这些方法通过利用剩余域上的解逐步构造全部解。然而，针对某些类型的代数方程，仅需部分解即可。本文结合两种现有方法，展示了如何利用有限链环上的Gröbner基求解有限交换环上的代数方程组。进而，通过引入斜多项式与Plücker坐标，论证了用于有限域上秩译码问题及MinRank问题的若干代数方法可推广至有限主理想环。