Let p(z) be a polynomial with leading coefficient 1, i.e. a monic polynomial. What is the maximum length of the curve
{z : |p(z)| = 1}
for each degree n?
Erdős conjectured that the maximum length was obtained by the polynomial p(z) = zn − 1. He was right, as Teng Zhang proved recently.
Examples
Here’s an example for degree five. The maximum length is obtained by p(z) = z5 − 1.
This curve has length 15.02.
Here’s the corresponding curve for p(z) = z5 − z
which has length 6.909.
In general the level set {z : |p(z)| = 1} need not be connected, in which case the length is the sum of the lengths of the components.
Here’s an example that tweaks the constant parameter of the optimal curve p(z) = z5 − 1.1.
This curve has (total) length 14.59.
Details
The length of the maximum curve for given n is the length for the polynomial p(z) = zn − 1. The only curves obtaining the maximum length are translations and rotations of this curve.
Note that for n = 1 the maximum curve is a circle and for n = 2 the curve is the lemniscate of Bernoulli. The general curve is called the Erdős lemniscate of order n, generalizing Bernoulli’s lemniscate.
The maximum length for given n is
21/n B(1/n, 1/2n)
where B is the beta function.
The Erdős lemniscates look similar to the “rose” curves r = cos(nθ), but these curves are not polynomial level sets.