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How can I ask Mathematica to find all polynomial $ f \! $s such that, for a constant $ k $:

$$ f(x) \ f(y) = f(x) + f(y) + f(xy) + k $$

I've solved it on paper and want to check my results. I've tried the following, none of which work:

  • Solve[f[x] f[y] == f[x] + f[y] + f[x y] + k, f]
  • Solve[f[x] f[y] == f[x] + f[y] + f[x y] + k, f[x]]
  • Solve[f[x] f[y] == f[x] + f[y] + f[x y] + k, f[z]]
  • Reduce[f[x] f[y] == f[x] + f[y] + f[x y] + k, f]
  • f[x_] := Sum[Subscript[c, i] x^i, {i, 0, n}]; Solve[f[x] f[y] == f[x] + f[y] + f[x y] + k, Subscript[c, 0]]
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  • $\begingroup$ Take a look at this quite analogous problem: Solving functional equations in Mathematica $\endgroup$
    – Artes
    Feb 20, 2020 at 2:20
  • $\begingroup$ I would consider polynomials of increasing degree, and write them in terms of their coefficients. Then solve for the coefficients $\endgroup$
    – mikado
    Feb 20, 2020 at 6:50

1 Answer 1

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It is better to use SolveAlways and to consider a specific value of n. Here is $n=4$:

f[x_] := Sum[Subscript[c, i] x^i, {i, 0, 4}]
{k, f[x]} /. SolveAlways[f[x] f[y] == f[x] + f[y] + f[x y] + k, {x, y}]
(* {{{-2, 1}, {-2, 1 + x^4}, {-2, 1 + x^3}, {-2, 1 + x^2}, {-2, 1 + x}} *)
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