Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and $J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ,\underline{x}^{\underline{b}_s}$ where $\underline{x}^{\underline{a}_i}=x_1^{a_{i1}}\cdots x_n^{a_{in}}$.

The Newton polynomial of $I$ is the set $$NP(I)=\{\underline{v}\in\mathbb{Q}^n_{\geq 0}|\underline{v}\geq \sum_{i=1}^s c_i\underline{a}_i,c_i\in\mathbb{Q}_{\geq 0},\sum_{i=1}^s c_i=1 \}.$$Similarly we define $NP(J)$, the Newton polygon of $J$.

$NP(I)$ and $NP(J)$ are closed convex sets in $\mathbb{Q}^n_{\geq 0}.$

Q) What is the relation between $NP(I^2J^3)$ and the sets $NP(I)$ and $NP(J).$

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and $J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ,\underline{x}^{\underline{b}_s}$ where $\underline{x}^{\underline{a}_i}=x_1^{a_{i1}}\cdots x_n^{a_{in}}$.

The Newton polyhedron of $I$ is the set $$NP(I)=\{\underline{v}\in\mathbb{Q}^n_{\geq 0}|\underline{v}\geq \sum_{i=1}^s c_i\underline{a}_i,c_i\in\mathbb{Q}_{\geq 0},\sum_{i=1}^s c_i=1 \}.$$ Similarly we define $NP(J)$, the Newton polyhedron of $J$.

$NP(I)$ and $NP(J)$ are closed convex sets in $\mathbb{Q}^n_{\geq 0}.$

Q) What is the relation between $NP(I^2J^3)$ and the sets $NP(I)$ and $NP(J).$