On a Lower-Bound for the Absolute Value of a Polynomial of Several Complex Variables Original Research Article

For an arbitrary polynomial P(z1, z2,...,zn) in complex space Cn we describe a set of nonnegative multi-indices α = (α1, α2,...,αn) such that for any n-tuple δ = (δ1,δ2,....,δn) ≥ 0 (where δj = 0 if αj = 0), one can find a system of "thin" sets Mj of widths ≤ to δj in directions of the axes zj, respectively, 1 ≤ j ≤ n, for which outside their union the absolute value of a polynomial is bounded away from zero by (δ/α)αΓα (Γα depends on P but not on δ). The prototype of this result is the well known Cartan′s Theorem on a lower bound for the modulus of a polynomial P(z), z ∈ C1.