to ${\mathbb{C}}^*$, $\mathbf{x}^{\mathbf{a}}$ is actually a character, i.e., a group homomorphism. In general, a system of Laurent monomials induced by $\mathbf{a}_1,\dots,\mathbf{a}_m \in {\mathbb{Z}}^n$ give rise to the group homomorphism $\mathbf{x} \mapsto (\mathbf{x}^{\mathbf{a}_1},\dots,\mathbf{x}^{\mathbf{a}_m})$ between ${({\mathbb{C}}^*)}^n$ and ${({\mathbb{C}}^*)}^m$. Of particular importance, is the case where $m=n$.

 \[lem:toric-automorphism\] Given vectors $\mathbf{a}_1,\dots,\mathbf{a}_n \in {\mathbb{Z}}^n$, the map $\mathbf{x} \mapsto (\mathbf{x}^{\mathbf{a}_1},\dots,\mathbf{x}^{\mathbf{a}_m})$ is an automorphism of ${({\mathbb{C}}^*)}^n$ if and only if $|\det [\mathbf{a}_1,\dots,\mathbf{a}_n]| = 1$ and in that case, the map is a bi-holomorphism.

For the integer matrix $A = [\mathbf{a}_1,\dots,\mathbf{a}_n]$ and $\mathbf{x} = (x_1,\dots,x_n)$ above, we use the compact notation ${\mathbf{x}}^A = (\mathbf{x}^{\mathbf{a}_1},\dots,\mathbf{x}^{\mathbf{a}_1})$ to represent the automorphism induced by $A$. Such a square integer matrix $A$ with $|\det(A)| = 1$ is said to be *unimodular*. More generally, an integer matrix (not necessarily square) is *totally unimodular* if all its nonsingular submatrices are unimodular. This concept also extend to lattice polytopes: A lattice simplex is *unimodular* if its normalized volume is 1, and a simplicial subdivision of a lattice polytope is *unimodular* if it consists of only unimodular simplices.

A *Laurent polynomial* is a finite linear combination of distinct Laurent monomials, i.e., an expression of the form $f = \sum_{\mathbf{a} \in S} c_{\mathbf{a}} \mathbf{x}^{\mathbf{a}}$ for some finite 