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I am trying to find the number of roots (or an upper bound thereof) of a very long transcendental equation. My question is what topic I should study in order to help.

I do not need to solve the equation, an upper bound (not infinity please!) for the number of roots would suffice.

PS, It cannot be converted into a polynomial.

Quadratic Reciprocity
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  • If you could share the equation you're interested in here, maybe someone could help with an explicit answer. In general, a good understanding of basic real analysis could help you figure and prove whether a solution exists, and if it does then whether it's unique or not, in the case of most simple equations. Try and also graph the equation - if done by computer, that could provide you with an informal answer you'd have to formally justify. –  Oct 07 '20 at 20:48
  • Thanks for the tips David Kipper, I have made some progress but need to know if $$a_1e^{f_1(x)}+a_2e^{f_2(x)}+....=0$$ has the same or fewer roots than the combined number of roots of $$f_1(x)=0 ,f_2(x)=0....$$ – Quadratic Reciprocity Oct 07 '20 at 22:16
  • What do you know about the number of roots of $f_i(x)?$ –  Oct 08 '20 at 13:44
  • That each f has less than 5 roots. – Quadratic Reciprocity Oct 09 '20 at 11:10

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