# GRevLex -- graded reverse lexicographical monomial order.

## Description

The graded reverse lexicographic order is defined by: $x^A > x^B$ if either $degree(x^A) > degree(x^B)$ or $degree(x^A) = degree(x^B)$ and the LAST non-zero entry of the vector of integers $A-B$ is NEGATIVE.

This is the default order in Macaulay2, in large part because it is often the most efficient order for use with Gröbner bases. By giving GRevLex a list of integers, one may change the definition of the order: $degree(x^A)$ is the dot product of $A$ with the argument of GRevLex.

 i1 : R = QQ[a..d]; i2 : a^3 + b^2 + b*c 3 2 o2 = a + b + b*c o2 : R i3 : S = QQ[a..d, MonomialOrder => GRevLex => {1,2,3,4}]; i4 : a^3 + b^2 + b*c 2 3 o4 = b*c + b + a o4 : S
The largest possible exponent of variables in the GRevLex order is 2^31-1. For efficiency reasons, it is sometimes useful to limit the size of monomials (this often makes computations more efficient).Use MonomialSize => 16, which allows maximal exponent 2^15-1, or MonomialSize => 8, which allows maximal exponent 2^7-1.
 i5 : B1 = QQ[a..d,MonomialSize=>16,MonomialOrder=>GRevLex]; i6 : B = QQ[a..d,MonomialSize=>16]; i7 : a^(2^15-1) 32767 o7 = a o7 : B i8 : C = QQ[a..d,MonomialSize=>8,MonomialOrder=>GRevLex]; i9 : try a^(2^15-1) else "failed" o9 = failed i10 : a^(2^7-1) 127 o10 = a o10 : C