The computation of Gröbner bases and standard bases is one of the core tools in symbolic computation and algebraic geometry, with important applications in polynomial ideal theory, singularity resolution, and other fields. In 2002, Professor Faugère first proposed the signature-based F5 algorithm, which introduced the concept of "signatures" to identify and discard unnecessary computations, significantly improving the algorithm's computational efficiency. In the study of signature-based Gröbner basis algorithms, the GVW algorithm serves as one of the important frameworks in this category. However, the Cover Theorem, which serves as the theoretical foundation of the GVW algorithm, has long been limited to global term orders. This limitation restricts the application of signature-based methods in broader algebraic structures, particularly in local rings and mixed order cases. This research fundamentally reconstructs the proof of the Cover Theorem by incorporating the core ideas of Mora's normal form algorithm, successfully extending it to arbitrary semigroup orders. This achievement expands the applicability of signature-based standard basis algorithms and provides a new theoretical foundation and practical tool for efficient symbolic computation in more general algebraic structures, such as local rings and mixed orders. This work was completed in collaboration with Dingkang Wang, Fanghui Xiao, and Xiaopeng Zheng.
Lu Dong is an Associate Professor and Master's Supervisor at Southwest Jiaotong University. He earned his Ph.D. from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and subsequently conducted postdoctoral research at the School of Mathematical Sciences, Beihang University. His research focuses on symbolic computation, primarily involving methods for solving polynomial systems, and the study of decomposition and equivalence problems of multivariate polynomial matrices. He has collaborated with scholars from both China and abroad to publish over ten research papers, and is supported by one project funded by the National Natural Science Foundation of China and one project funded by the Natural Science Foundation of Sichuan Province.
