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I'm doing my senior capstone on Schanuel's conjecture and in my final presentation I wanted to discuss why this conjecture is important. I have found tons of applications in field theory and proving other conjectures. I have found slightly more concrete applications in proving algebraic independence and roots of exponential polynomials, but I'm not sure where these could be used in real world problems (or if they could be at all). Does anybody have any recommendations of papers which discuss this or do you know of any applications? I've done a lot of research and read a lot of papers on this, but I haven't come up with anything "real world". Thank you in advance for your time and knowledge

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    $\begingroup$ I think trying to find "real world" applications is not a good use of your time. Does "real world" mean some area of applied mathematics, something in physics, something in a physical science, something involving physical science but not basic scientific research (e.g. an engineering field), something involving a natural science topic (e.g. biology, ecology, biochemistry, environmental science, geology, etc.), something involving a natural science OR a social science (e.g. economics, psychology, political theory, etc.), something outside of a science-type field, . . .? $\endgroup$ Apr 2, 2021 at 15:15
  • $\begingroup$ That said, there might be applications involved with non-commensurable periods of periodic phenomena in chaos theory type stuff (e.g. orbits will be dense, and so the "system" eventually comes arbitrarily close infinitely often to any possible configuration). $\endgroup$ Apr 2, 2021 at 15:20
  • $\begingroup$ Thank you for your comment. I was thinking of something just a little more concrete than super abstract field theory. I enjoy abstract mathematics but I thought a question that may come up is "where is this applicable?" I will look into any applications in chaos theory. Thank you again. $\endgroup$ Apr 2, 2021 at 18:01
  • $\begingroup$ Possibly googling things like "ergodic", "irrational rotation", "dense", etc. will lead to something (google search for all three of these terms). Some early "real world" precursors to these notions was the problem of the Solar System's stability over long periods of time (note this is an $n$-body problem for $n \geq 3),$ which motivated some of Poincare's work which led to other things (e.g. see vicinity of p. 335 and p. 339 of this 1915 survey paper). $\endgroup$ Apr 2, 2021 at 18:16
  • $\begingroup$ en.m.wikipedia.org/wiki/Schanuel%27s_conjecture $\endgroup$ Apr 9, 2021 at 12:40

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The topic of Schanuel's conjecture is algebraic independence. Applications of the conjecture are e.g. in abstract algebra, algebraic geometry and number theory.

It's a difficult task to look for concrete applications of Schanuel's conjecture outside of mathematics. Undeniably, pure mathematics has an impact on applied mathematics, and applied mathematics has an impact on its applications outside of mathematics.

Schanuel's conjecture is used e.g. for determining existence, type and number of solutions of exponential polynomial equations and for determining of exceptional points of functions (i.e. algebraic values at algebraic points). Maybe this is important for calculation of these solutions.

The elementary functions are compositions of algebraic functions, $\exp$ and/or $\ln$.
Schanuel's conjecture can be used i.a. for deciding if an equation of elementary functions has solutions in the elementary numbers (means in terms of elementary functions), one kind of closed-form numbers.

This is used by itself, but i.a. also for deciding if a given elementary function can have partial inverses that are elementary functions.

Solutions in closed form can give e.g. hints for calculating the solutions numerically or to discover relationships - inside and outside of mathematics.

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

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Not an answer - a suggested frame change.

Rather than look for a forced or artificial "application" to justify this part of abstract mathematics, focus instead on its importance purely as mathematics.

You don't learn language only to read the newspaper. It gives you access to poetry. You don't learn mathematics only for its practical applications. You can preempt the "where is this useful?" question by framing your presentation as about the beauty of abstraction for its own sake.

PS. I just noticed that this is a two year old question. The presentation has happened. I hope it went well.

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