multidegree monoids and lattices

For any element $f=\sum _{I,J} f_{I J}\xi^I \eta^J \in A_n(k)$, we denote by $\operatorname{multideg}(f)$ the multidegree of $f$. That means,

$$\operatorname{multideg}(f) =\max \{ (I,J) \in \mathbb{N}^{2n} ; f_{I J} \neq 0\},$$

where we employ the lexicographic order on $\mathbb{N}^{2n}$. For any subalgebra $A$ of $A_n(k)$, we define its multidegree monoid $\operatorname{multideg}(A)$ as follows.

$$\operatorname{multideg}(A)=\{\operatorname{multideg}(f); f\in A\}.$$

It is a sub monoid of $\mathbb{N}^{2n}$. We use several norms for indices in $\mathbb{N}^{2n}$. Among them is the ``$\ell^1$-norm'' $\vert t_1\vert+\dots+\vert t_{2n}\vert$. of $(t_1,\dots,t_{2n})$ The total degree $\operatorname{totaldeg}(f)$ of an element $f$ of $A_n(k)$ is then defined to be the $\ell^1$-norm of $\operatorname{multideg}(f)$. By a total degree of a derivation or an algebra endomorphism of $A_n(k)$ we mean the maximum of the total degree of the image of the standard generators $\xi_1,\dots,\xi_n,\eta_1,\dots,\eta_n$.

Definition 2   For any subset $S$ of $\mathbb{N}^{2n}$, we denote by $S_{\leq d}$ the set of all elements of $S$ whose $\ell^1$-norms are less than or equal to $d$. We denote by $\operatorname{{}^aHF}$ "the affine Hilbert function" of $S$, that is,

$$\operatorname{{}^aHF}_S(i)=\char93 (S_{\leq i}) \qquad ( i \in \mathbb{N})$$

For any subalgebra $A$ of $A_n(k)$, we define its affine Hilbert function $\operatorname{{}^aHF}_A$ as the affine Hilbert function $\operatorname{{}^aHF}_{\operatorname{multideg}{A}}$ of multidegree of $A$.

Lemma 18   Let $m$ be a positive integer. Let $S$ be a sub monoid of $\mathbb{N}^m$. Let $L=\mathbb{Z}S$ be the submodule of $\mathbb{Z}^n$ generated by $S$. Then the following conditions are equivalent.

1. There exists a positive real number $\epsilon$ such that $\operatorname{{}^aHF}_S(i) \geq \epsilon i^{m}$ for all $i»0$.
2. $\operatorname{rank}(L)=m$

$(2)\implies (1)$: Take $I_1,I_2,\dots,I_m\in S$ which are linearly independent over $\mathbb{Q}$. Put $c=\max ( \vert\vert I_1\vert\vert _{\ell^1}, \vert\vert I_2\vert\vert _{\ell^1},\dots, \vert\vert I_n\vert\vert _{\ell^1})$. Then a map $\alpha$ defined by

$$\alpha:\mathbb{N}^m \ni (a_1,\dots,a_m)\mapsto a_1 I_1+a_2 I_2+\dots + a_m I_m \in S$$

is injective and satisfies $\alpha (\mathbb{N}^m_{\leq d})\subset S_{\leq c d}$ for every positive integer $d$. Thus we have

$$\operatorname{{}^aHF}_S(d)\geq \binom{m+[d/c]}{m} \geq \frac{d^m}{(c+1)^m m!}$$

when $d$ is large enough. ($[\bullet]$ denotes the Gaussian symbol.)

$(1)\implies (2)$: Assume on the contrary that $r=\operatorname{rank}(L)<m$. Then the module $L$, being torsion free, is isomorphic to $\mathbb{Z}^r$. Let $I_1,\dots,I_r$ be elements of $S$ which forms a $\mathbb{Z}$-basis of $L$. Then a map

$$\beta: \mathbb{R}^r\ni (t_1,t_2,\dots,t_r) \mapsto (t_1 I_1+ t_2 I_2 +\dots + t_r I_r)\in \mathbb{R}^m$$

is an injective linear map from $\mathbb{R}^r$ to $\mathbb{R}^m$. Thus we may easily see that there exists a real number $M$ such that

$$\vert\vert\sum_{i=1}^r t_i I_i \vert\vert _{\ell^1}<1 \implies \vert\vert(t_1,\dots,t_r)\vert\vert _{\ell^1}<M$$

This implies that $\char93 S_{<d}$ is smaller than the number of elements of $\mathbb{Z}^r$ which are shorter (in $\ell^1$-norm) than $M d$. $\qedsymbol$

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