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.

翻译：可求长路径的签名是张量代数中的一个张量级数，其系数为路径的确定迭代积分。签名刻画了路径在重参数化的广义形式下的等价类。陈省身（K. T. Chen）的经典结果表明，对数签名（签名的对数）是一个李级数。若李级数具有有限次数，则其为多项式形式。本文证明：对数是多项式的充要条件是路径在重参数化下为直线。因此，可求长路径的对数签名要么次数为1，要么具有无限支撑。尽管本结论针对可求长路径，证明过程使用了粗糙路径理论的结论，特别是签名在重参数化下刻画粗糙路径的经典性质。