 
Summary: ON HERMITIAN POLYNOMIAL OPTIMIZATION
MIHAI PUTINAR
Abstract. We compare three levels of algebraic certificates for eval
uating the maximum modulus of a complex analytic polynomial, on a
compact semialgebraic set. They are obtained as translations of some
recently discovered inequalities in operator theory. Although they can
be stated in purely algebraic terms, the only known proofs for these
decompositions have a transcendental character.
1. Introduction
Let z = (z1, ..., zd) be the complex coordinates in Cd. Then real coordi
nates of the underlying space R2d are denoted by x = (x1, ..., x2d), where
zk = xk +ixd+k. We will work in the polynomial algebra R[x], and consider
there the convex hulls:
2
= co{p2
; p R[x]},
and
2
h = co{q2
; q C[z]}.
