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What is the easiest or fastest way to extract the homogeneous part of a polynomial in Mathematica. For instance, if there were a function homog_part[f,n], I could use it to extract the second, first and zeroth degree homogeneous part of $f=x^2+y^2+xy+x+y+7$ by writing homog_part[f,2] to get $x^2+y^2+xy$, homog_part[f,1] to get $x+y$ and homog_part[f,0] to get $7$.

I was thinking of using Series, but here you would give individual degrees of each variable while for the homogeneous part you need the total degree.

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    $\begingroup$ A question like this really should be accompanied by test input in cut-and-pasteable form. $\endgroup$
    – Daniel Lichtblau
    Commented Oct 29, 2017 at 14:48
  • $\begingroup$ @Daniel. Apologies. I agree. Some of the answer have these input so I will not add this in the question anymore. But I will keep this in mind for future reference/questions. $\endgroup$
    – quantum
    Commented May 16, 2018 at 14:15

3 Answers 3

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f[x_, y_] = x^2 + y^2 + x y + x + y + 7;
homoPart = Total[Pick[MonomialList[#], Total[CoefficientRules[#][[All, 1]], {2}], #2]] &;
homoPart[f[x, y], #] & /@ Range[0, 3]

{7, x + y, x^2 + x y + y^2, 0}

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SeriesCoefficient is useful for this. We just multiply all variables by a new variable and extract a degree term in that.

homogeneousPart[poly_, vars_, deg_] := 
 Module[{t}, 
  SeriesCoefficient[poly /. Thread[vars -> t*vars], {t, 0, deg}] /. 
   t -> 1]

Example:

pol = x^2 + y^2 + x*y + x + y + 7;

Table[homogeneousPart[pol, {x, y}, j], {j, 3, 0, -1}]

(* Out[6]= {0, x^2 + x y + y^2, x + y, 7} *)
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  • $\begingroup$ +1 so neat. Nicer than all the manipulation I did with CoefficientRules :) $\endgroup$
    – ubpdqn
    Commented Oct 30, 2017 at 6:51
  • $\begingroup$ This is closely related to Jens's solution here. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Oct 31, 2017 at 5:56
  • $\begingroup$ @J.M. Thanks for the link. I see that I upvoted that one. Should have realized this has come up before. $\endgroup$
    – Daniel Lichtblau
    Commented Oct 31, 2017 at 14:13
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hp[f_, n_] := 
 Total@Cases[{Plus @@ #1, #2 Inner[Power, Variables[f], #1, Times, 
        1]} & @@@ CoefficientRules[f], {n, w_} :> w]

e.g. hp[f[x,y],#]&/Range[0,2] yields {7, x + y, x^2 + x y + y^2}

or hp[x^3 + 3 x^2 y + 2, #] & /@ Range[0, 3] yields {2, 0, 0, x^3 + 3 x^2 y}

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