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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 $NP(J).$

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    $\begingroup$ $NP(I)$ is the convex hull (in $\mathbb Q^n$) of all of the monomials in $I$. On the other hand monomials in the product of two ideals $I$ and $J$ are exactly products of a monomial in $I$ and a monomial in $J$, So its convexhull is the minkowski sum of Newton polytopes of $I$ and $J$. In particular we have: $NP(I^2J^3) = 2NP(I)+3NP(J)$. $\endgroup$
    – Mostafa - Free Palestine
    Commented May 20, 2015 at 7:59

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