Kernel of map between polynomial rings that takes monomials to monomials

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.

• Those things are called, usually, binomial relations. Sep 5 '11 at 21:28
• OK, I changed it to that term (which I suppose is a little more descriptive). Sep 6 '11 at 1:32
• 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).
– M P
Sep 6 '11 at 9:35