Banach lattices of homogeneous polynomials not containing $c_0$

First we develop a technique to construct Banach lattices of homogeneous polynomials. We obtain, in particular, conditions for the linear spans of all positive compact and weakly compact $n$-homogeneous polynomials between the Banach lattices $E$ and $F$, denoted by ${\cal P}_{\cal K}^r(^n E; F)$ and $\mathcal{P}_{\mathcal{W}}^r(^n E; F)$, to be Banach lattices with the polynomial regular norm. Next we study when the following are equivalent for ${\cal I} = {\cal K}$ or ${\cal I} = {\cal W}$: (1) The space $\mathcal{P}^r(^n E; F)$ of regular polynomials contains no copy of $c_0$. (2) ${\cal P}_{\mathcal{I}}^r(^n E; F)$ contains no copy of $c_0$. (3) ${\cal P}_{\mathcal{I}}^r(^n E; F)$ is a projection band in $\mathcal{P}^r(^n E; F)$. (4) Every positive polynomial in $\mathcal{P}^r(^n E; F)$ belongs to ${\cal P}_{\cal I}^r(^nE;F)$. The result we obtain in the compact case can be regarded as a lattice polynomial Kalton theorem. Most of our results and examples are new even in the linear case $n = 1$.