= Divergent algebraic root in a singular perturbation
{title2=$\lambda\sim\varepsilon^{-p}\mu$}
= Divergent perturbation root
{synonym}
A polynomial whose degree decreases when its parameter vanishes may have roots escaping to infinity. Such a <singular perturbation> root is found by balancing powers after rescaling $\lambda=\varepsilon^{-p}\mu$. If the root enters an exponential, even a lower-order additive root correction can matter on a fixed interval when its leading exponential grows. This phenomenon is different from a repeated root of a fixed polynomial.
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