Inreal algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set UinRn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set KinU, there exist positive constants α and C such that, for all xinK
Here α can be large.
The following form of this inequality is often seen in more analytic contexts: with the same assumptions on f, for every p ∈ U there is a possibly smaller open neighborhood Wofp and constants θ ∈ (0,1) and c > 0 such that
A special case of the Łojasiewicz inequality, due to Polyak [ru], is commonly used to prove linear convergenceofgradient descent algorithms.[1]
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