An Attempt to Enhance Buchberger's Algorithm by Using Remainder Sequences and GCD Operation

This paper proposes a new method of enhancing Buchberger's algorithm for the lexicographic order (LEX-order) Groebner bases. The idea is to append polynomials to the input system, where we generate polynomials to be appended by computing PRSs (polynomial remainder sequences) of input polynomials and making them close to basis elements by the GCD operation. In order to do so, we first restrict the input system to satisfy three reasonable conditions (we call such systems "healthy"), and we compute redundant PRSs so that the variable-eliminated remainders are not in a triangular form but in a rectangular form; we call the corresponding PRSs "rectangular PRSs (rectPRSs)"; see 2. A for details of rectPRSs. The lowest order element of the LEX-order Groebner basis can be computed from the last set of remainders of rectPRSs and the GCD operation; see [20]. We generate other polynomials to be appended by computing rectPRSs of leading coefficients of mutually similar remainders and converting the order-reduced leading coefficient to a polynomial in the given ideal. By these we are able to compute the second-lowest element of the Groebner basis, too. The research is on-going now. Our method seems promising but it contains many problems, theoretical as well as computational, so we open our research.