Find where a function does not depend on one of its argument

Question

Say I have a function $f(x,y)$ with $x$ and $y$ in some domains. How do I find the subdomain in $y$ for which $f$ does not depend on $x$.

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My special case one could consider as an example:

I have a Spherical Harmonics $Y_{l,m}(\theta,\phi)$. I want to find constraints on $l,m,\theta$ that would imply a non-dependence upon $\phi$. I know this independence because of physical reasons Mathematica could not infer.

How can I tell Mathematica something like:

$\qquad \mathsf{Solve}[Y_{l,m}(\theta,\phi)==Y_{l,m}(\theta,\phi^\prime)\ \forall\phi,\phi^\prime\in[0,2\pi]]$

I know the answer in this case:

$\qquad Y_{l,m}(\theta,\phi)=Y_{l,m}(\theta,\phi^\prime)\ \forall\phi,\phi^\prime\in[0,2\pi] \rightarrow m=0$

Note: Mathematica knows SphericalHarmonicY[l, m, theta, phi].

In reference to a comment by @patrick-stevens: one could differentiate the function. Nevertheless, it is not the case in general and I am looking for other methods.

Answer 1

In general it would boil down to doing something like:

f[x_, y_] := Sin[y] x

Resolve[ForAll[x, f[x, y] == f[0, y]], Reals]
(* Sin[y] == 0 *)

or

f[x_, y_] := 2 + 9 x - 6 x y + x y^2

Resolve[ForAll[x, f[x, y] == f[0, y]], Reals]
(* -3 + y == 0 *)

However, quite often this will lead to systems that Mathematica can't solve, like your case

g[l_, m_, θ_, ϕ_] := SphericalHarmonicY[l, m, θ, ϕ]

Resolve[ForAll[ϕ, 0 <= ϕ <= 2 π, g[l, m, θ, ϕ] == g[l, m, θ, 0]]]

although FindInstance can find one special case:

FindInstance[Resolve[ForAll[ϕ, 0 <= ϕ <= 2 π, g[l, m, θ, ϕ] == g[l, m, θ, 0]]], 
    {l, m, θ}]
(* {{l -> 0, m -> 0, θ -> 0}} *)