Enter the textit{dynamic} algorithm, which computes a basis textit{and} an ordering textit{simultaneously}, using a ``tentative'' Hilbert function to estimate efficiency. Unfortunately, record-keeping quickly grows very large, outweighing benefts even for “toy ideals”.

This talk, based on joint work with Massimo Caboara of the University of Pisa, introduces dynamic algorithms, describes two effective techniques to minimize record-keeping, and reports on experiments with other criteria for efficiency.