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 ${\mathcal{P}}_{\mathcal{K}}^r{(}^{n}E;F)$ 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 $\mathcal{I}=\mathcal{K}$ or $\mathcal{I}=\mathcal{W}$: (1) The space ${\mathcal{P}}^r{(}^{n}E;F)$ of regular polynomials contains no copy of $c_0$. (2) ${\mathcal{P}}_{\mathcal{I}}^r{(}^{n}E;F)$ contains no copy of $c_0$. (3) ${\mathcal{P}}_{\mathcal{I}}^r{(}^{n}E;F)$ is a projection band in ${\mathcal{P}}^r{(}^{n}E;F)$. (4) Every positive polynomial in ${\mathcal{P}}^r{(}^{n}E;F)$ belongs to ${\mathcal{P}}_{\mathcal{I}}^r{(}^{n}E;F)$. 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 $n=1$.