X_k\ \text{for some}\ k.$$ An algebraic set is usually represented by its generating polynomials in practice. Note that an algebraic set may have different sets of generating polynomials. However, by defining $$I(X)\triangleq\{f\in{\mathbb{K}}[x]\mid\forall {\pmb{x}}\in X.\, f({\pmb{x}})=0\}$$ for any $X\subseteq {\mathbb{K}}^d$, one can easily verify that $I(Z(P))$ is the maximal set that generates $Z(P)$. So, any algebraic set $X$ can be identified by the ideal $I(X)$. The membership $f\in I(Z(P))$ for any polynomial $f$ and any finite set $P=\{f_1,\cdots,f_n\}$ of polynomials is equivalent to the unsatisfiability of $f_1=0\wedge f_2=0\wedge\cdots\wedge f_n=0\wedge f\neq0$, which is decidable [@Tarski51].

Additionally, noting that $X_1\subset X_2\Leftrightarrow I(X_1)\supset I(X_2)$ for two algebraic sets $X_1$ and $X_2$, it follows from Theorem \[thm:asc-chain\] that

\[thm:des-chain\]For any descending chain of algebraic sets $$X_1\supseteq X_2\supseteq\cdots\supseteq X_n\supseteq\cdots$$ of ${\mathbb{K}}^d$, there exists an $N\in\mathbb{N}$ such that $X_n=X_N$ for all $n\geq N$.

Monomial ideals and Hilbert’s functions
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An ideal $I$ is called monomial if it can be generated by a set of monomials. A monomial ideal always has a basis $\{m_1,m_2,\dots,m_s\}$ of monomials (due to Dickson’s Lemma), and any monomial $m\in \langle m_1,m_2,\cdots,m_s\rangle$ should be a multiple of some $m_i$.

For a monomial ideal $I\subseteq{\mathbb{K}}[{\pmb{x}}]$, a function $H_I:{\mathbb{N}}\rightarrow{\mathbb{N}}$ is defined as $$H_I(n)=\dim_{\mathbb{K}}{\mathbb{K}}_n[{\pmb{x}}]/I_n=\dim_{\mathbb{K}}{\mathbb{K}}_n[{\pmb{x}}]-\dim_{\mathbb{K}}I_n,$$ where ${\mathbb{K}}_n[{\pmb{x}}]$ is 