Summary: We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of [\textit{V. Guruswami} and \textit{M. Sudan}, ibid. 45, No. 6, 1757-1767 (1999; Zbl 0958.94036); \textit{M. A. Shokrollahi} and \textit{H. Wasserman}, ibid. 45, No. 2, 432-437 (1999; Zbl 0947.94024)] to run in polynomial time. We do this by presenting a root-finding algorithm for univariate polynomials over function fields when their coefficients lie in finite-dimensional linear spaces, and proving that there is a polynomial size representation, given which the root-finding algorithm runs in polynomial time.