# Polynomials in nature

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.

Edit:

There are some formulas under beam deflection category.

I am particularly interested in full polynomials, as opposed to pure power laws. For example $$y=at^2+v_0 t + y_0$$ for a projectile uses a complete polynomial, while Boltzman law is just a power law.

• 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. Commented Oct 9, 2013 at 17:51
• The volume of the sun is a degree-three polynomial of its radius...
– user856
Commented Oct 9, 2013 at 17:55
• The ideal gas law can be thought of as an equality between a degree-three polynomial and a degree-two polynomial. Commented Oct 9, 2013 at 18:04
• I've always been intrigued by the $r^4$ occuring in the Hagen-Poiseuille law. Commented Oct 9, 2013 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. Commented Oct 9, 2013 at 18:07

This approach is justified by the Stone-Weierstrass Theorem, which states that continuous functions can be uniformly approximated by a polynomial. The natural quantities we observe can typically be assumed to be continuous, at least to some degree, so this approach applies. If the true function can be represented by a convergent power series, then higher degree polynomials will make better approximations **. For instance, consider the set of discrete observations of a time series, $$\big((1,\sin1),\ (2,\sin2),\ (3,\sin3),\ldots,\ (n,\sin n)\big)$$. Then, a good approximation of the time series would be $$\sin t$$, which could be used to interpolate and extrapolate for all real values. Since $$\sin t$$ can be given by a power series, in this case higher degree polynomials are indeed better approximations to the data set. However, in practice, it is typically better to approximate a function with just a sufficiently high degree polynomial. Using arbitrarily high degree polynomials takes longer to compute and does not always make a better approximation; Runge's phenomenon shows that raising the degree of interpolating polynomials can sometimes increase approximation error.