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What polynomials occur in "nature"? I am interested in polynomials of degree three and higher. I am aware of Stefan Boltzmann Law and Chemical Equilibrium Examples.

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Bruno Buchberger once told me something along the lines of "you can solve your problem if you can express it in polynomials." (Heavily paraphrased, highly innacurate, long-time-ago memory: please, nobody read very far into it :) ) That was right after a talk about automated proofs, so I imagine he was thinking along the lines of encoding proofs in polynomials. Much later, I saw a really cool automated proof of the concurrence of medians in a triangle using Buchberger's algorithm. – rschwieb Oct 9 '13 at 17:51
The volume of the sun is a degree-three polynomial of its radius... – Rahul Oct 9 '13 at 17:55
The ideal gas law can be thought of as an equality between a degree-three polynomial and a degree-two polynomial. – vadim123 Oct 9 '13 at 18:04
I've always been intrigued by the $r^4$ occuring in the Hagen-Poiseuille law. – Hagen von Eitzen Oct 9 '13 at 18:04
1st and 2nd orders very common but it is true that from 3 onwards it gets harder. I could generalise the answer above to any volumes but it's a rather small family still. Perhaps you could use them for approximations using Taylor series for example getting rid of $x^n$ onwards for some $n>3$ but it would only be an approximation of nature. Kepler's third law uses a third degree relationship. One more idea that come to my mind would be describing dynamical systems, could it be animals interacting or anything else really. – user88595 Oct 9 '13 at 18:07

Power Laws are probably what you are looking for and are very, very prevalent in physics, economics, linguistics and well, nature. They occur for phenomena that exhibit scale invariance. All forms of exponents occur. The M-sigma law has a power of 4 for example.

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