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My son accidentally discovered and, given $f(x)=-x^{2}-1x+1$, $f\left(x\right)^{f\left(x\right)^{f\left(x\right)^{f\left(x\right)}}}$ intersects 1.0 (i.e., $y=1.0$) at the negative of the golden ratio, i.e., $-1.618$ (and at the golden ratio -1 on the right side of the $y$ axis, i.e., $0.618$), and that adding EVEN numbers of repeated powers of $f(x)$ improves the precision (unsurprisingly). He discovered this just by playing around with graphs on desmos. I'd love to be able to explain to him why this is, and I'm pretty good at moderately advanced algebra, but I don't know what it means to raise a polynomial to the power of another polynomial! There's obviously something fairly simple going on, because by monkeying with the coefficients of the base polynomial you can get various versions of integer roots, and since the golden ration is basically a version of $\sqrt{5}$, I wasn't too surprised to see it there, but ... well, what does raising a polynomial to a polynomial power mean? Thanks!

Xander Henderson
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jackisquizzical
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  • That the sequence of powers converges is, ultimately, a consequence of the Banach fixed point theorem. The algebra required to show that the power tower converges to $-\phi$ is a little more complicated, but (I think) should be relatively elementary. – Xander Henderson Apr 20 '21 at 14:46
  • A distinct, but related post: https://math.stackexchange.com/questions/2468073/ . – Xander Henderson Apr 20 '21 at 14:48
  • The expression $f(x)^{f(x)}$ in the context of "playing around with graphs on [D]esmos" suggests that one is evaluating the function $f(x)$ at values $x$ which give positive values. Not every expression "Raising a Polynomial to a Polynomial Power" is going to make sense, and I would be skeptical of how the Desmos software handles arguments $x$ where $f(x)$ is not positive. – hardmath Apr 20 '21 at 14:58
  • @hardmath: You are correct. In fact, Desmose stops plotting altogether at the point where f(x) is negative. – jackisquizzical Apr 20 '21 at 17:06
  • BTW, how DO you raise a polynomial (or, for that matter, anything) to a polynomial power??? – jackisquizzical Apr 20 '21 at 19:58

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