Let $X: (\Omega, \mathcal F, \mathbb P) \to (\mathbb R, \mathcal B(\mathbb R))$ be a real-valued random variable, and set
\begin{align*}
D_X := \{t \in \mathbb R : \mathbb E[e^{tX}] < \infty\}.
\end{align*}
The moment generating function of $X$ is the map
\begin{align*}
M_X: D_X \to (0,\infty).
\end{align*}
For $t \in D_X$, it is defined by
\begin{align*}
M_X(t):=\mathbb E[e^{tX}].
\end{align*}
Let $k$ be a field and let $X\subset k^n$ be an affine algebraic set. Let
\begin{align*}
I(X)=\{f\in k[x_1,\ldots,x_n]\mid f(a)=0 \text{ for every } a\in X\}.
\end{align*}
The coordinate ring of $X$ over $k$ is the quotient ring
\begin{align*}
k[X]=k[x_1,\ldots,x_n]/I(X).
\end{align*}
An abelian group is a set $G$ equipped with a binary operation
\begin{align*}
\cdot:G\times G\to G
\end{align*}
such that $(G,\cdot)$ is a group and
\begin{align*}
gh = hg
\end{align*}
for all $g,h \in G$.
Let $E$ be a set, let $(Y,d_Y)$ be a metric space, and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n:E \to Y$. The sequence $(f_n)_{n=1}^{\infty}$ is uniformly Cauchy on $E$ if for every $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that, for all $m,n \ge N$ and all $x \in E$,
\begin{align*}
d_Y(f_n(x),f_m(x))<\varepsilon.
\end{align*}
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and let $S$ be a countable set. A Markov chain on $S$ is a sequence $(X_n)_{n \ge 0}$ of random variables $X_n : \Omega \to S$ such that, for every $n \ge 0$ and every $i_0, \ldots, i_{n+1} \in S$ with
\begin{align*}
\mathbb P(X_0 = i_0, \ldots, X_n = i_n) > 0,
\end{align*}
one has
\begin{align*}
\mathbb P(X_{n+1} = i_{n+1} \mid X_0 = i_0, \ldots, X_n = i_n) = \mathbb P(X_{n+1} = i_{n+1} \mid X_n = i_n).
\end{align*}
Let $E \subset \mathbb R$, and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n: E \to \mathbb R$. The sequence $(f_n)$ is uniformly bounded on $E$ if there exists $M \ge 0$ such that
\begin{align*} |f_n(x)| \le M \end{align*}
for every $n \in \mathbb N$ and every $x \in E$.
The set of natural numbers is the set $\mathbb{N}$ whose elements are generated from a distinguished element $1$ by repeated application of a successor operation $S: \mathbb{N} \to \mathbb{N}$, subject to the Peano axioms: $1 \in \mathbb{N}$; $S(n) \in \mathbb{N}$ for every $n \in \mathbb{N}$; $S(n) \ne 1$ for every $n \in \mathbb{N}$; $S(m)=S(n) \implies m=n$ for all $m,n \in \mathbb{N}$; and if $A \subset \mathbb{N}$, $1 \in A$, and $n \in A \implies S(n) \in A$, then $A=\mathbb{N}$.
Let $k$ be a field and let $n \in \mathbb{N}$. An affine variety over $k$ is a subset $X \subset \mathbb{A}^n_k := k^n$ for which there exists a set $S \subset k[x_1, \ldots, x_n]$ such that
\begin{align*}
X = \{p \in \mathbb{A}^n_k : f(p)=0 \text{ for every } f \in S\}.
\end{align*}
Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X: \Omega \to \{0,1,2,\dots\}$ be a random variable. Write
\begin{align*}
p_X(k)=\mathbb P(X=k), \qquad k\in \{0,1,2,\dots\}.
\end{align*}
The probability generating function of $X$ is the function $G_X:\{z\in\mathbb C:|z|\le1\}\to\mathbb C$ defined by
\begin{align*}
G_X(z)=\mathbb E[z^X]=\sum_{k=0}^{\infty}p_X(k)z^k.
\end{align*}
Let $k$ be a field. In the classical set-theoretic convention used on this page, a projective variety over $k$ is a subset $X \subset \mathbb{P}^n_k(k)$ of the form
\begin{align*}
X = V_+(S)(k)
\end{align*}
for some $n \in \mathbb{N}$ and some set $S \subset k[x_0, \ldots, x_n]$ of homogeneous polynomials, where $V_+(S)$ denotes their common projective zero locus.
Let $V$ be a vector space over a field $k$, and let $U\subset V$ be a linear subspace. For $v\in V$, write
\begin{align*}
v+U:=\{v+u:u\in U\}.
\end{align*}
The quotient space of $V$ by $U$ is the set
\begin{align*}
V/U:=\{v+U:v\in V\},
\end{align*}
equipped with the operations
\begin{align*}
+:(V/U)\times(V/U)\to V/U,\qquad (v+U,w+U)\mapsto (v+w)+U,
\end{align*}
\begin{align*}
\cdot:k\times(V/U)\to V/U,\qquad (\lambda,v+U)\mapsto (\lambda v)+U.
\end{align*}
