An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences (2505.10246v2)

Abstract: 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, particularly for large-scale systems. These results indicate that our algorithm can serve as an effective pre-processing tool for accelerating Groebner basis computations in generic polynomial sequence solving.