On uniqueness of extension of homogeneous polynomials

We study the uniqueness of norm-preserving extension of n-homogeneous polynomials in Banach spaces. We show that norm-preserving extensions of n-homogeneous polynomials do not need to be unique for n ≥ 2 in real Banach spaces, and for n ≥ 3 in a large class of complex Banach function spaces. We find further a geometric condition, which in particular yields that a unit ball in X does not possess any complex extreme point, under which for every norm-attaining 2-homogeneous polynomial on a complex symmetric sequence space X there exists a unique norm-preserving extension from X to its bidual X**. In particular, if m_ψ is a Marcinkiewicz sequence space and m_ψ⁰, is its subspace of order continuous elements, we show that every norm-attaining 2-homogeneous polynomial on m _ψ⁰ has a unique norm-preserving extension to its bidual m_ψ if and only if no element of a unit ball of m _ψ is a complex extreme point. We then apply these results to obtain some necessary conditions for the uniqueness of extension of 2-homogeneous polynomials from a complex symmetric space X to its bidual X**.