= Square-root splitting of a double polynomial root
{title2=$x-x_0=O(\epsilon^{1/2})$}
Suppose a perturbed <polynomial> has the local form $A(x-x_0)^2+\epsilon B+O((x-x_0)^3,\epsilon(x-x_0),\epsilon^2)$ with $AB\ne0$. <Dominant balance> gives $x-x_0=\epsilon^{1/2}z$ and $Az^2+B=0$. The two resulting <Puiseux series> split a <double root>; their leading displacements can be real or a <complex conjugate> pair. Ordinary <Taylor series> in $\epsilon$ cannot describe this splitting.
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