Computation of H-Basis and Syzygies via QR Decomposition

H-bases provide an attractive alternative to Gröbner bases for the study of polynomial ideals because they are independent of monomial orderings and therefore preserve structural properties that may be obscured in Gröbner basis computations. However, the practical use of H-bases has been limited by the lack of efficient algorithms for their construction, particularly due to the difficulty of computing syzygy modules and eliminating redundant generators. In this paper, we present a new algorithm for computing syzygies and H-bases based on numerical linear algebraic techniques. The proposed approach utilizes QR decomposition to construct syzygies directly from coefficient matrices, avoiding the need for monomial orderings and Gröbner basis computations. A recursive framework derived from QR factorization is developed to compute both syzygy modules and H-bases while simultaneously defining an associated reduction process. In contrast to existing methods based on Gröbner bases, Schreyer’s theorem, or singular value decomposition, the proposed algorithm requires fewer matrix reductions and relies on computationally less demanding operations. Furthermore, its termination criterion is determined by an upper bound independent of Gröbner basis theory. Consequently, the method provides a more streamlined and conceptually simpler framework for the computation of syzygies and H-bases of polynomial ideals.