Giving a polynomial, say

a x^2 + b x y + c y^2

MonomialList[a x^2 + b x y + c y^2, {x, y}] just gives

{a x^2, b x y, c y^2}

How can I get a list without the coefficients? I mean, the following list

{x^2, x y, y^2}

The motivation of this question is, I am only interested in the structure of the polynomial itself, i.e., which kinds of of monomials are there, while their explicit coefficient are not relevant.


3 Answers 3


You can generate the monomials by using CoefficientRules, like this

In[55]:= monomialList[poly_, vars_] := Times @@ (vars^#) & /@ CoefficientRules[poly, vars][[All, 1]]
         monomialList[a x^2 + b x y + c y^2, {x, y}]

Out[56]= {x^2, x y, y^2}

A pattern matching method:

fn[x_, {var__}] := List @@ Pick[x, x, Alternatives[var]^_.]

fn[a x^2 + b x y + c y^2, {x, y}]
{x^2, x y, y^2}

But a better approach I believe is (hopefully now corrected at last):

fn2[x_, var_] := Collect[List @@ Expand @ x, var, 1 &]

fn2[a x^2 + b x y + c y^2, {x, y}]
{x^2, x y, y^2}
fn2[x (x^2 + y^2), {x, y}]
{x^3, x y^2}
fn2[p x + a x^2 + b x y, {x, y}]
{x, x^2, x y}
  • $\begingroup$ Thank you so much. It is very clever and efficient! $\endgroup$
    – panda.G
    Jul 21, 2015 at 16:43
  • $\begingroup$ @user29373 Don't miss my last update; I like it best. And you're welcome. $\endgroup$
    – Mr.Wizard
    Jul 21, 2015 at 16:44
  • $\begingroup$ Thanks again @Mr.Wizard! That is really fantastic. $\endgroup$
    – panda.G
    Jul 21, 2015 at 16:54
  • 1
    $\begingroup$ Maybe add Expand in there? Try fn2[x(x^2 + y^2)] as it is now. $\endgroup$ Jul 21, 2015 at 17:02
  • $\begingroup$ @Marius Thank you. I think the simple change I just made catches that case. Would you please test it? $\endgroup$
    – Mr.Wizard
    Jul 21, 2015 at 17:07

This uses some undocumented functionality:

poly = a x^2 + b x y + c y^2; vars = {x, y};
dl = GroebnerBasis`DistributedTermsList[poly, vars];
Inner[Power, vars, #, Times] & /@ dl[[1, All, 1]]

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