Built-in Mathematica Symbol

MonomialList

MonomialList[poly]
gives the list of all monomials in the polynomial poly.

MonomialList[poly, {x₁, x₂, ...}]
gives the list of monomials with respect to the variables x_i in poly.

MonomialList[poly, {x₁, x₂, ...}, order]
puts the monomials in the specified order.

Possible settings for order are "Lexicographic", "DegreeLexicographic", "DegreeReverseLexicographic", "NegativeLexicographic", "NegativeDegreeLexicographic", "NegativeDegreeReverseLexicographic" or an explicit weight matrix.

Monomials are sorted on the basis of their exponent vectors with respect to the variables x_i.

"NegativeLexicographic" corresponds to applying Sort to the list of exponent vectors.

"Lexicographic" gives the reverse of "NegativeLexicographic", and is the default for MonomialList.

"DegreeLexicographic" sorts first with respect to total degree, then by using the ordering defined by "Lexicographic".

"DegreeReverseLexicographic" sorts first with respect to total degree, then in the negative lexicographic order by starting from the last variable.

"NegativeDegreeLexicographic" and "NegativeDegreeReverseLexicographic" sort from lower to higher total degree.

An explicit weight matrix w defines an ordering given by "Lexicographic" ordering of the w.v_i, where the v_i are the exponent vectors.

MonomialList[poly, All, order] is equivalent to MonomialList[poly, Variables[poly], order].

Get the list of monomials:

Use "DegreeLexicographic" monomial ordering:

Specify the same ordering using weight matrix:

Reduce the coefficients modulo 2:

Plus or Total reconstructs the original polynomial:

CoefficientRules gives a different representation:

Obtain "NegativeDegreeReverseLexicographic" from "DegreeLexicographic":

The list given by Variables[poly] is not always sorted: