Lower Bounds on the Continuation of Holomorphic Functions

We show that continuation of holomorphic functions needs time at least Ω(n2) in the uniform case, which is optimal up to a polylogarithmic factor. In the non-uniform case, i.e. assuming f to be computable in time O(t) on an open subset of its domain, we show, that continuation of f cannot be robustly done in time o(n⋅t(n)). This again is optimal up to a polylogarithmic factor.