Signature-based algorithm under non-compatible term orders and its application to change of ordering

The notion of the compatibility between a term order in a polynomial ring R and a module term order in $R^l$ is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.