  We propose a new formalism to analyse the extremum structure of
scale-invariant effective potentials. The problem is stated in a compact matrix
form, used to derive general expressions for the stationary point equation and
the mass matrix of a multi-field RG-improved effective potential. Our method
improves on (but is not limited to) the Gildener-Weinberg approximation and
identifies a set of conditions that signal the presence of a radiative minimum.
When the conditions are satisfied at different scales, or in different
subspaces of the field space, the effective potential has more than one
radiative minimum. We illustrate the method through simple examples and study
in detail a Standard-Model-like scenario where the potential admits two
radiative minima. Whereas we mostly concentrate on biquadratic potentials, our
results carry over to the general case by using tensor algebra.
