[Way too long for a comment but not quite an answer.]
We can use numerical integration at modest precision to show that the correct number of solutions, counting by multiplicity, cannot exceed 143.
First thing is to observe that roots satisfy the univariate x - Sin[6*Pi*Sin[6*Pi*z]] == 0
. Since the Sin[...]
, for real x
, cannot exceed 1 in absolute value, this means all roots are bounded by -+1. A simple computation shows that the endpoints of this interval are not themselves roots.
that I use the argument principal and integrate around a thin box where top and bottom are at y=+-epsilon (I set epsilon to 1/1000) and left and right edges are at -1-epsilon and 1+epsilon respectively. I could in theory use -+1 but I wanted to allow a bit of extra room because it looks like there are roots nearby. What the integral shows is that there are 143 roots inside this box. Coupled with the fact that you can find 143 explicitly real roots, this shows that is all there can be.
ee = z - Sin[6*Pi*Sin[6*Pi*z]];
eps = 1/1000;
bottom =
NIntegrate[D[ee, z]/ee, {z, -1 - eps - eps*I, 1 + eps - eps*I},
WorkingPrecision -> 100, MaxRecursion -> 16]
right = NIntegrate[D[ee, z]/ee, {z, 1 + eps - eps*I, 1 + eps + eps*I},
WorkingPrecision -> 100, MaxRecursion -> 16]
top = NIntegrate[D[ee, z]/ee, {z, 1 + eps + eps*I, -1 - eps + eps*I},
WorkingPrecision -> 100, MaxRecursion -> 16]
left = NIntegrate[
D[ee, z]/ee, {z, -1 - eps + eps*I, -1 - eps - eps*I},
WorkingPrecision -> 100, MaxRecursion -> 16]
(* Out[27]= 0.*10^-122 +
450.23417218584244411148838277077827524114697014604184527662050714016\
98412748656717060712169899295271 I
-0.9864227225020110113303789618093628009517460354017128772034304401520\
951759377849022622369660067864026 I
0.*10^-122 +
450.23417218584244411148838277077827524114697014604184527662050714016\
98412748656717060712169899295271 I
-0.9864227225020110113303789618093628009517460354017128772034304401520\
951759377849022622369660067864026 I *)
(bottom + right + top + left)/(2*Pi*I)
(* 143.000000000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000 + 0.*10^-122 I *)
I remark that there are probably complex-valued roots lurking nearby. When I set epsilon to 1/100 I had a slightly higher count.
In[41]:= Timing[ Length[NSolve[{x == Sin[6 Pi y], y == Sin[6 Pi x]}, {x, y}, Reals]]] Out[41]= {0.180000, 143}
$\endgroup$Plot[Evaluate@{Sin[6π x], Table[{ArcSin[x]/(6π) + k/3, -ArcSin[x]/(6π) + 1/6 - k/3}, {k, -3, 3, 1}]}, {x, -1, 1}, AspectRatio -> Automatic]
. It seems easy to count 12x11+2x6 solutions. $\endgroup$