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CoefficientRules

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`CoefficientRules`

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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.

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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].

Examples

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Basic Examples (1)Summary of the most common use cases

Get exponents and coefficients of monomials:

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-ctcwzy

Out[1]=1

Scope (1)Survey of the scope of standard use cases

Use "DegreeReverseLexicographic" monomial ordering:

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-b8csnn

Out[1]=1

Specify the same ordering using weight matrix:

In[2]:=2

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https://wolfram.com/xid/0b8cn0k4y6-doqecd

Out[2]=2

Options (1)Common values & functionality for each option

Modulus (1)

Reduce the coefficients modulo 2:

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-hlm66i

Out[1]=1

Properties & Relations (2)Properties of the function, and connections to other functions

FromCoefficientRules reconstructs the original polynomial:

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-lspjtq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0b8cn0k4y6-kocky6

Out[2]=2

MonomialList gives a different representation:

In[3]:=3

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https://wolfram.com/xid/0b8cn0k4y6-i83krh

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-bof22u

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0b8cn0k4y6-jo9s9

Out[2]=2

Possible Issues (1)Common pitfalls and unexpected behavior

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

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-e4vsy3

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0b8cn0k4y6-bz064l

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0b8cn0k4y6-dlgf70

Out[3]=3

Neat Examples (2)Surprising or curious use cases

Visualize monomial orderings in 2D:

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-dplr7j

The standard built-in orderings:

In[2]:=2

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https://wolfram.com/xid/0b8cn0k4y6-hcdq2g

In 2D some orderings cannot be distinguished:

In[3]:=3

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https://wolfram.com/xid/0b8cn0k4y6-gb745u

Out[3]=3

Visualize monomial orderings in 3D:

In[1]:=1

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https://wolfram.com/xid/0b8cn0k4y6-f7l40x

In[2]:=2

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https://wolfram.com/xid/0b8cn0k4y6-yk14d

In 3D all orderings are distinct:

In[3]:=3

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https://wolfram.com/xid/0b8cn0k4y6-gdxl7x

Out[3]=3

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