Subtotal ordering -- a pedagogically advantageous algorithm for   computing total degree reverse lexicographic order
  Total degree reverse lexicographic order is currently generally regarded as most often fastest for computing Groebner bases. This article describes an alternate less mysterious algorithm for computing this order using exponent subtotals and describes why it should be very nearly the same speed the traditional algorithm, all other things being equal. However, experimental evidence suggests that subtotal order is actually slightly faster for the Mathematica Groebner basis implementation more often than not. This is probably because the weight vectors associated with the natural subtotal weight matrix and with the usual total degree reverse lexicographic weight matrix are different, and Mathematica also uses those the corresponding weight vectors to help select successive S polynomials and divisor polynomials: Those selection heuristics appear to work slightly better more often with subtotal weight vectors.   However, the most important advantage of exponent subtotals is pedagogical. It is easier to understand than the total degree reverse lexicographic algorithm, and it is more evident why the resulting order is often the fastest known order for computing Groebner bases.   Keywords: Term order, Total degree reverse lexicographic, tdeg, grevlex, Groebner basis 
David R. Stoutemyer
cs.SC cs.MS