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 xi.
CoefficientRules [poly,{x1,x2,…},order]
gives the result with the monomial ordering specified by order.
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 xi.
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
open all close allBasic Examples (1)
Get exponents and coefficients of monomials:
Scope (1)
Use "DegreeReverseLexicographic" monomial ordering:
Specify the same ordering using weight matrix:
Options (1)
Modulus (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:
Tech Notes
Related Guides
History
Text
Wolfram Research (2008), CoefficientRules, Wolfram Language function, https://reference.wolfram.com/language/ref/CoefficientRules.html.
CMS
Wolfram Language. 2008. "CoefficientRules." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CoefficientRules.html.
APA
Wolfram Language. (2008). CoefficientRules. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoefficientRules.html
BibTeX
@misc{reference.wolfram_2025_coefficientrules, author="Wolfram Research", title="{CoefficientRules}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/CoefficientRules.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_coefficientrules, organization={Wolfram Research}, title={CoefficientRules}, year={2008}, url={https://reference.wolfram.com/language/ref/CoefficientRules.html}, note=[Accessed: 17-November-2025]}