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Is it possible to write a finite equation consisting ONLY of exponentials, logs and $x^n$ where $n$ is any real number which has an infinite number of non-equal real roots ? (trivial examples like $x-x=0$, $(\sqrt(x)^2)$/$x =1$, imaginary numbers, piecewise functions to be excluded). For instance can an equation like the one below (or a bigger one) have an inf number of real roots? I think not but how does one go about proving it ?

$\frac{exp(x^3-4x).ln(\sqrt(x^x-cosh(2x+1))+1}{exp(2x+1)}$ + $\frac{6.exp(x^3-4x).ln(\sqrt(4x^x-asinh(2-6x))+1}{x^3-exp(9x+1)}$ + $\frac{exp(x^5-4x.ln(\sqrt(8x^x-cosh(24x-16))-9}{564exp(2x-91)}$ - $\frac{2x^6-13x^2+xln(\sqrt(x^2-atanh(29x-1))-1}{x^7-2x^3+9x^6-2x+ln(1+x+x^8)}$ = $(ln(x)^3-1/x)^x$

I am interested in hearing of a general argument and not one specifically for the equation above.

ps: only the functions I have mentioned in line 1 are allowed, all other functions, operators (exc + - / x), integrals, derivatives are disallowed.

Quadratica
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1 Answers1

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The first order theory $\Bbb{R}_{\mathsf{exp}}$ over the language that adds the exponential function $\mathsf{exp}(\cdot)$ to the usual field operators on the reals is know to be o-minimal, which implies that the set of roots of an equation of the sort you are interested in is a finite union of points and intervals (so if there are infinitely many distinct roots they are all contained in a finite set of intervals each of whose points is a root). See the Wikipedia article on exponential fields for references.

Rob Arthan
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  • It seems you are not excluding the possibility of inf roots ? Surely, no such equation can have inf roots...whether on an open or closed interval ? – Quadratica May 14 '19 at 21:28
  • @Quadratica: what about $e^{\log((x+1)^2)/2}+e^{\log((x-1)^2)/2}-2=0$ ? –  May 14 '19 at 21:35
  • It does not have inf roots.....I tried it with log and ln. – Quadratica May 14 '19 at 21:44
  • I think Yves meant $e^{\log((x+1)^2)/2}-e^{\log((x-1)^2)/2}-2=0$. The point is that what you call "trivial examples" are tricky to characterise. – Rob Arthan May 14 '19 at 21:54
  • I included the word trivial to exclude those equations which when plotted show an inf number of roots but not an inf number of turning points. – Quadratica May 14 '19 at 22:06
  • An infinite number of turning points is impossible by o-minimality. – Rob Arthan May 14 '19 at 22:09
  • Right, is that so. Then I guess that answers my Q. Please direct me to further info and if possible give me an outline of the reasoning which shows that inf number of turning points are impossible. I have a physics degree but math is my passion and I will need to dig deep into this stuff to figure it. – Quadratica May 14 '19 at 22:20
  • I would recommend Lou van den Dries's book *Tame Topology and O-minimal Structures" as a starting point. – Rob Arthan May 14 '19 at 22:26
  • Thank you, just had a look at some stuff about 0-minimality etc and it will take some working on!! I hope you're right about the turning points!! – Quadratica May 14 '19 at 22:38