= Reduced local defining functions differ by a holomorphic unit
{title2=$g=uf,\quad u(p)\ne0$}
The <local ring> of a <complex manifold> is the <unique factorization domain> of convergent power series in local coordinates. A reduced <local defining function of a complex analytic hypersurface> has each local irreducible factor once. The analytic zero-set ideal is the radical of its <principal ideal>, and reducedness makes that <principal ideal> radical. Thus two reduced equations for the same germ generate the same ideal and differ by a unit. A globally irreducible hypersurface may have several local branches; reducedness includes each once. Without reducedness, $f$ and $f^2$ have the same zero set and need not differ by a unit.
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