TITLE: Kernel of map between polynomial rings that takes monomials to monomials
QUESTION [11 upvotes]: Let $k$ be a field (say of characteristic $0$).  Let $z_1,\ldots,z_n \in k[y_1,\ldots,y_m]$ be monomials, and consider the ring homomorphism $\phi : k[x_1,\ldots,x_n] \rightarrow k[y_1,\ldots,y_m]$ defined by $\phi(x_i)=z_i$.  There are many elements in the kernel of $\phi$ of the form $x_1^{a_1} \cdots x_n^{a_n} - x_1^{b_1} \cdots x_n^{b_n}$ for nonnegative integers $a_i$ and $b_i$.  These correspond to "relations between the $z_i$".  Call these relations the "binomial relations".  My questions is whether the kernel of $\phi$ is generated by the binomial relations.
Here is an example to clarify the above.  Set $n=4$ and $m=2$.  Define our function $\phi : k[x_1,x_2,x_3,x_4] \rightarrow k[y_1,y_2]$ by
$$\phi(x_1)=y_1^2 y_2$$
$$\phi(x_2)=y_1 y_2^2$$
$$\phi(x_3)=y_2^3$$
$$\phi(x_4)=y_1^3$$
One can then easily check that the kernel of $\phi$ is generated by the set $\{x_1^2 - x_2 x_4, x_2^2 - x_1 x_3, x_1 x_2 - x_3 x_4\}$, which of course is a collection of binomial relations.

REPLY [3 votes]: Posting my comment as an answer.
There is a proof of this fact in the book "Toric varieties" by Cox-Little-Schenck (and presumably in many other places!): Proposition 1.1.9 of the online version of the book (that may no longer be available).