Loewner’s “forgotten” theorem

Yesterday I discovered an instructive paper on “Loewner’s forgotten theorem” by Peter Albers and Serge Tabachnikov on the arXiv. I don’t know in how far the result has really been forgotten, it was published in the Annals in 1948. Fair enough, it has only 4 citations on MathSciNet.

So what is Loewner’s theorem? It says that all rotation numbers of a certain family of smooth closed curves are non-negative. The curves covered by the theorem can be constructed as follows. Take two real monic polynomials P and Q, where P has degree n+1 and Q has degree n. Assume that P and Q have only real roots \(a_0, a_1, \dots, a_n\) and \(b_1, b_2, \dots, b_n\) which are distinct and interlaced \[a_0 < b_1 < a_1 < b_2 < a_2 < \dots < b_n < a_n\]. Let f be a smooth periodic function. Then

\[ \gamma(t) = (P(d/dt)(f), Q(d/dt)(f)) \]

is a smooth closed curve. Loewner’s theorem says that \(\gamma\) is of non-negative circulation, i.e. every point x not on \(\gamma\) has a positive rotation number. The proof is rather instructive and can be found in both papers linked above.