CoefficientRules

CoefficientRules[poly,{x₁,x₂,…}]

gives the list {{e₁₁,e₁₂,…}c₁,{e₂₁,…}c₂,…} of exponent vectors and coefficients for the monomials in poly with respect to the x_i.

CoefficientRules[poly,{x₁,x₂,…},order]

gives the result with the monomial ordering specified by order.

Details and Options

CoefficientRules works whether or not poly is explicitly given in expanded form.
CoefficientRules[poly] is equivalent to CoefficientRules[poly,Variables[poly]].
Possible settings for order are the same as in MonomialList.
The default order is "Lexicographic".
CoefficientRules[poly,vars,Modulus ->m] computes the coefficients modulo m.
CoefficientRules[poly,All,order] is the same as CoefficientRules[poly,Variables[poly],order].

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Get exponents and coefficients of monomials:

Use "DegreeReverseLexicographic" monomial ordering:

Specify the same ordering using weight matrix:

Reduce the coefficients modulo 2:

FromCoefficientRules reconstructs the original polynomial:

MonomialList gives a different representation:

For two variables "DegreeLexicographic" and "DegreeReverseLexicographic" coincide:

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

Visualize monomial orderings in 2D:

The standard built-in orderings:

In 2D some orderings cannot be distinguished:

Visualize monomial orderings in 3D:

In 3D all orderings are distinct:

Introduced in 2008

(7.0)

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