We present a quasi maximum principle stating roughly that holomorphic solutions of a given partial differential equation with constant coefficents in C-n, P(C) u = 0, (dagger) achieve essentially their maximal growth on a certain algebraic hypersurface Gamma related to the operator. We prove it in the case where P is homogeneous and Gamma is the conjugate dual cone, and also in the case where P(D) = D-1(2) + ... + D-n(2) and Gamma is the complexified real sphere. We obtain a weak (semi-local) variant of the quasi maximum principle for certain non-homogeneous operators P(D), in which case Gamma is the conjugate dual cone related to the principal part of the operator. This weaker variant is closely intertwined with several other notions. One of them is a quasi balayage principle for solutions of (dagger), involving the ''sweeping'' of measures in C-n onto Gamma. (C) 1997 Academic Press. [References: 16]
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