We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties conjectured to hold for generic sequences-specifically, that their leading monomial ideals are weakly reverse lexicographic and that their Hilbert series follow a known closed-form expression. The algorithm incrementally constructs the set of leading monomials degree by degree by comparing Hilbert functions of monomial ideals with the expected Hilbert series of the input ideal. To enhance computational efficiency, we introduce several optimization techniques that progressively narrow the search space and reduce the number of divisibility checks required at each step. We also refine the loop termination condition using degree bounds, thereby avoiding unnecessary recomputation of Hilbert series. Experimental results confirm that the proposed method substantially reduces both computation time and memory usage compared to conventional Groebner basis computations for computing the leading monomials of a minimal Groebner basis of generic sequences.

翻译：本文提出一种高效算法，用于计算齐次多项式一般序列的极小Gröbner基的首项单项式。该方法通过利用一般序列所满足的结构性质（具体而言，其首项单项式理想具有弱逆字典序性质，且其希尔伯特级数遵循已知闭式表达式），规避了代价高昂的多项式约化过程。该算法通过逐次比较单项式理想的希尔伯特函数与输入理想的预期希尔伯特级数，逐步按次数构建首项单项式集合。为提升计算效率，我们引入了多种优化技术：逐步缩小搜索空间、减少每一步所需的整除性检验次数，并通过次数界改进循环终止条件以避免不必要的希尔伯特级数重计算。实验结果表明，与传统Gröbner基计算方法相比，本方法在计算一般序列极小Gröbner基的首项单项式时，能显著降低计算时间与内存占用。