On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of $n$-homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space $X$ that admits contractive projection of finite rank of at least dimension 2, for every $n\ge 3$ there exists an $n$-homogeneous polynomial on $X$ that has infinitely many extensions to $X^{**}$. We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice $E$, for instance when $E$ satisfies an upper $p$-estimate with constant one for some $p>2$, any $2$-homogeneous polynomial on $E$ attaining its norm at $x\in E$ with a finite rank band projection $P_x$, has a unique extension to its bidual $E^{**}$. We apply these results in a class of Orlicz sequence spaces.