The reverse lexicographic order is defined by: x
A > x
B if the LAST non-zero entry of the vector of integers
A-B is NEGATIVE. This is a local order, not a global order. Therefore Groebner bases over this ring only give generators over the local ring whose fractions are all elements not in the ideal generated by the variables.
i1 : R = QQ[a..d];
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i2 : a^3 + b^2 + b*c + a*c^2 + b^2*c
3 2 2 2
o2 = a + b c + a*c + b + b*c
o2 : R
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Computations of Groebner bases for local orders are done using Mora's algorithm.