Smooth globally PL functions on contractible manifolds are nonlinear sums of squares f = f* + ||φ||² via a submersion φ, yielding a Euclidean-or-exotic dichotomy for minimizers and a convexifying metric change.
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Smooth, globally Polyak-{\L}ojasiewicz functions are nonlinear least-squares
Smooth globally PL functions on contractible manifolds are nonlinear sums of squares f = f* + ||φ||² via a submersion φ, yielding a Euclidean-or-exotic dichotomy for minimizers and a convexifying metric change.