Extracting elements from a List in a specific way

Question

I have a set of polynomial expressions in multiple variables:

U = x[1] x[2] + x[1] x[3] + x[2] x[3]
F = m1^2 x[1]^2 x[2] + m2^2 x[1] x[2]^2 + m1^2 x[1]^2 x[3] + m1^2 x[1] x[2] x[3] + m2^2 x[1] x[2] x[3] + m3^2 x[1] x[2] x[3] - p^2 x[1] x[2] x[3] + m2^2 x[2]^2 x[3] + m3^2 x[1] x[3]^2 + m3^2 x[2] x[3]^2
G = U + F

The relevant variables in this case (and for all of my examples) are the various "x[i]"-s, where "i" is an integer.

To split the terms in G, I use this answer.

ClearAll[toList]
toList = # /. {_[a__] :> {a}, a_?AtomQ :> {a}} &;
listresG = toList[G];

What I need: A way to extract the terms appearing in U and F from the expression of G. I find doing this non-trivial since Mathematica sum does some kind of monomial ordering of its own that might change the place of each term. For eg, in this example, the explicit expression for G happens to be

G = x[1] x[2] + m1^2 x[1]^2 x[2] + m2^2 x[1] x[2]^2 + x[1] x[3] + m1^2 x[1]^2 x[3] + x[2] x[3] + m1^2 x[1] x[2] x[3] + m2^2 x[1] x[2] x[3] + m3^2 x[1] x[2] x[3] - p^2 x[1] x[2] x[3] + m2^2 x[2]^2 x[3] + m3^2 x[1] x[3]^2 + m3^2 x[2] x[3]^2

which has clearly altered the positioning of the terms relative to where they appear in U and F.

I have tried doing Cases[listresG, x[_Integer]*x[_Integer]] to extract the terms appearing in U from G, but this doesn't work.

Answer 1

UTerms = List @@ U
(*{x[1] x[2], x[1] x[3], x[2] x[3]}*)

FTerms = List @@ F
(*{m1^2 x[1]^2 x[2], m2^2 x[1] x[2]^2, m1^2 x[1]^2 x[3], 
 m1^2 x[1] x[2] x[3], m2^2 x[1] x[2] x[3], 
 m3^2 x[1] x[2] x[3], -p^2 x[1] x[2] x[3], m2^2 x[2]^2 x[3], 
 m3^2 x[1] x[3]^2, m3^2 x[2] x[3]^2}*)

GTerms = List @@ G
(*{x[1] x[2], m1^2 x[1]^2 x[2], m2^2 x[1] x[2]^2, x[1] x[3], 
 m1^2 x[1]^2 x[3], x[2] x[3], m1^2 x[1] x[2] x[3], 
 m2^2 x[1] x[2] x[3], m3^2 x[1] x[2] x[3], -p^2 x[1] x[2] x[3], 
 m2^2 x[2]^2 x[3], m3^2 x[1] x[3]^2, m3^2 x[2] x[3]^2}*)

I"m not 100% clear what you are meaning by "terms". If you mean terms in the sum then:

UinG = Intersection[Gterms, UTerms]
(*{x[1] x[2], x[1] x[3], x[2] x[3]}*)

FinG = Intersection[Gterms, FTerms]
(*{m1^2 x[1]^2 x[2], m2^2 x[1] x[2]^2, m1^2 x[1]^2 x[3], 
 m1^2 x[1] x[2] x[3], m2^2 x[1] x[2] x[3], 
 m3^2 x[1] x[2] x[3], -p^2 x[1] x[2] x[3], m2^2 x[2]^2 x[3], 
 m3^2 x[1] x[3]^2, m3^2 x[2] x[3]^2}*)

If that's not what you meant, let me know in a comment.

Answer 2

THIS IS AN EXTENDED COMMENT RATHER THAN AN ANSWER

To amplify upon MarcoB's comment about indexed variables:

Clear["Global`*"]

To format specified indexed variables (e.g., x and m) as subscripts use

(Format[#[n_]] := Subscript[#, n]) & /@ {x, m};

U = x[1] x[2] + x[1] x[3] + x[2] x[3]

F = (-p^2 + m[1]^2 + m[2]^2 + m[3]^2) x[1] x[2] x[3] + 
  m[3]^2 (x[1] + x[2]) x[3]^2 + m[2]^2 x[2]^2 (x[1] + x[3]) + 
  m[1]^2 x[1]^2 (x[2] + x[3])

G = x[1] x[2] + x[1] x[3] + 
  x[2] x[3] + (-p^2 + m[1]^2 + m[2]^2 + m[3]^2) x[1] x[2] x[3] + 
  m[3]^2 (x[1] + x[2]) x[3]^2 + m[2]^2 x[2]^2 (x[1] + x[3]) + 
  m[1]^2 x[1]^2 (x[2] + x[3])

Answer 3

To gather terms from U and F in G, you may first transform the polynomials into lists and then use "GatherBy" checking for membership in e.g. U. This gives a one liner:

GatherBy[List @@ G, MemberQ[List @@ U, #] &]