Show that the number of points of $V(I)$ is at most $m_1m_2...m_n$ if $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$.

Question

This is an exercise from Ideals, Varieties and Algorithms by Cox et al.

Let $I\subset \mathbb{C}[x_1,...,x_n]$ be an ideal such that for each $i$, some power $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$. Show that the number of points of $V(I)$ is at most $m_1m_2...m_n$.

The notation $\left\langle \text{LT}(I) \right\rangle$ means the ideal generated by the leading terms of polynomials in $I$.

Hint: Use the following two facts:

1. $k[x_1,...,x_n]/I$ is isomorphic as a $k$-vector space to $S=\text{span}(x^{\alpha}: x^{\alpha}\notin \left\langle \text{LT}(I) \right\rangle)$;

2. Let $G$ be a Groebner basis for $I$. Then with the assumption in the problem, we know that there exists some $k_i\leq m_i$ such that $x_i^{k_i}=\text{LM}(g_i)$ for some $g_i\in G$.

My attempt:

The monomials in $S$ could only be $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ where $\alpha_i<m_i$.

Consider $[x_i^j]$ in $\mathbb{C}[x_1,...,x_n]/I$ where $j=0,1,...,m_i$. They have to be linearly dependent.(I know this is wrong now.) So $$\sum^{m_i}_{j=0}c_j[x_i^j]=\left[\sum^{m_i}_{j=0}c_jx_i^j\right]=[0]$$

Thus

$$\sum^{m_i}_{j=0}c_jx_i^j\in I$$

Since this is a polynomial with degree $m_i$ in $\mathbb{C}[x_i]$, it has at most $m_i$ roots. This is true for each coordinate, so $V(I)$ contains at most $m_1\cdots m_n$ points.

Question:

This was my first attempt. Then I realized that $[x_i^j]$ do not have to be linearly dependent. One of them could depend on some other monomials with other variables. I still think I need to end up with one variable function and use the linear dependence to show the number of roots is bounded by $m_i$. But I couldn't find them. Or is there another approach?

Thank you very much for any help!

Answer 1

A rough sketch of the proof in spoilers:

1)

Show that $V(I)$ is finite, e.g. $V(I)=\{p_1,\ldots,p_m\}$

2)

Derive the kronecker functions $\delta_{p_i}$ on $V(I)$ as polynomials $f_i$

3)

Show that $f_i$ are linear independent in $\mathbb{C}[x_1,\ldots,x_n]/I$