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：我们提出了一种高效算法，用于计算齐次多项式一般序列的极小Gröbner基的首项单项式。我们的方法通过利用针对一般序列所推测的结构性质（具体而言，其首项单项式理想是弱反向字典序的，且其希尔伯特级数遵循已知的封闭形式表达式）来规避代价高昂的多项式约简。该算法通过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，按度数递增逐步构造首项单项式集合。为提升计算效率，我们引入了若干优化技术，这些技术能逐步缩小搜索空间并减少每一步所需的整除性检查次数。我们还利用度数边界改进了循环终止条件，从而避免不必要的希尔伯特级数重复计算。实验结果表明，与传统的Gröbner基计算方法相比，所提方法在计算一般序列的极小Gröbner基的首项单项式时，能显著降低计算时间和内存占用。