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This is a question on a proof of Theorem 1.2.7 in Sturmfel's Algorithms in Invariant Theory. I will restate the whole proof up to where I am confused.

First some notation. Let $\sigma_i(x_1, \ldots, x_n)$ be the $i$-th elementary symmetric polynomial. Let $h_i(x_1, \ldots,x_n)$ be the $i$-th complete symmetric polynomial. Let $I$ be the ideal of $\mathbb{C}[\overline{x}, \overline{y}] = \mathbb{C}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ generated by $\sigma_i(x_1, \ldots, x_n) - y_i$.

Next I will state the following useful theorem without proof:

$\textbf{Theorem 1.2.6:}$ Let $I$ be any ideal of $\mathbb{C}[\overline{x}]$ and fix any monomial order. Then the residue classes of standard monomials (monomials that are not in the initial ideal of $I$, $\text{init}(I)$) form a $\mathbb{C}$ vector space basis of $\mathbb{C}[\overline{c}] / I$.

$\textbf{Theorem 1.2.7:}$ The unique reduced Groebner basis of $I$ with respect to the lexicographic monoimial order induced by $x_1 > x_2 > \cdots > x_n > y_1 > y_2 > \cdots > y_n$ is $$G = \left\{ h_k(x_k, \ldots, x_n) + \sum_{i = 1}^l (-1)^i h_{k - i}(x_k, \ldots, x_n) \cdot y_i \mid k = 1, \ldots, n \right\}.$$

$\textbf{Proof}$: First, note that for all $k = 1, \ldots, n$ that $$h_k(x_k, \ldots, x_n) + \sum_{i = 1}^l (-1)^i h_{k - i}(x_k, \ldots, x_n) \cdot \sigma_i(x_1, \ldots, x_n) = 0.$$ This, $G \subseteq I$.

Now grade $\mathbb{C}[\overline{x}, \overline{y}]$ by $\deg(x_i) = 1$ and $\deg(y_i) = i)$. Thus, $I$ is homogenous and $R = \mathbb{C}[\overline{x}, \overline{y}] / I$ is isomorphic as a graded algebra to $\mathbb{C}[\overline{x}]$. Thus, the Hilbert Series of $R$ is $\frac{1}{(1 - z)^{-n}}$. By $\textbf{Theorem 1.2.6}$ we have that the Hilbert series of $\mathbb{C}[\overline{x}, \overline{y}] / \text{init}(I)$ is also $\frac{1}{(1 - z)^{-n}}$.

Now consider the ideal $J = \langle x_1, x_2^2, \ldots, x_n^n\rangle$. This is the initial ideal of $G$. Clearly $J \subseteq \text{init}(I)$. (And now this is where I start getting confused) Our assertion states that these two ideals are equal. For the proof, it is sufficient to verify that the Hilbert series of $R' = \mathbb{C}[\overline{x}, \overline{y}] / J$ is equal to the Hilbert series of $R$.

So here are my questions

1: When the proof says "Our assertion states that these two ideals are equal" what assertion are they referring to?

2: More importantly, why is it sufficient to verify that these Hilbert series are equivalent?

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  • $\begingroup$ You absolutely cannot claim this, for instance $I=(x)$ and $J=(y)$ in $\Bbb C[x,y]$. You need to provide more context about the problem you're trying to solve in an edit to this post. Please type the claim that's giving you trouble instead of linking the book or using a picture. $\endgroup$ Mar 31 at 21:45
  • $\begingroup$ I am sorry for the poor earlier post. I have (hopefully sufficiently) clarified my question $\endgroup$
    – Anakin Dey
    Apr 1 at 0:34

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  1. The "assertion" is the statement of the Theorem, which states that $G$ is a Gröbner basis for $I$. By definition, this is saying that the leading terms of the polynomials in $G$ generate the initial ideal of $I$.

  2. If they were not equal, then since they are both homogeneous, there is some degree $d$ in which they differ; since $J_d$ is always a subspace of $init(I)_d$ (since $J \subseteq init(I)$), we conclude that $J_d$ is a proper subspace of $init(I)_d$. In particular, the Hilbert series would not be the same. This is a generalization of the statement that if $V \subseteq W$ are both finite-dimensional subspaces of the same vector space and their dimensions agree, then they are equal.

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