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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.

MORE INFORMATION

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].

EXAMPLES

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Basic Examples (1)

Get exponents and coefficients of monomials:

In[1]:=

Out[1]=

Scope (1)

Use "DegreeReverseLexicographic" monomial ordering:

Specify the same ordering using weight matrix:

Options (1)

Reduce the coefficients modulo 2:

Properties & Relations (2)

FromCoefficientRules reconstructs the original polynomial:

MonomialList gives a different representation:

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

Possible Issues (1)

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

Neat Examples (2)

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: