  The signature of a rectifiable path is a tensor series in the tensor algebra
whose coefficients are definite iterated integrals of the path. The signature
characterises the path up to a generalised form of reparametrisation. It is a
classical result of K. T. Chen that the log-signature (the logarithm of the
signature) is a Lie series. A Lie series is polynomial if it has finite degree.
We show that the log-signature is polynomial if and only if the path is a
straight line up to reparametrisation. Consequently, the log-signature of a
rectifiable path either has degree one or infinite support. Though our result
pertains to rectifiable paths, the proof uses results from rough path theory,
in particular that the signature characterises a rough path up to
reparametrisation.
