\documentclass{article}
\usepackage{iftex}
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\begin{document}
$x$
\pagebreak

$y$
\pagebreak

$z$
\pagebreak

$t$
\pagebreak

$\ideal{x^2, y^2, x*z, y*z}$
\pagebreak

$\ideal{x, y}$
\pagebreak

$\ideal{x, y, z}$
\pagebreak

$\ideal{x*y*z^3, x^2y^2}$
\pagebreak

$\ideal{x*y*z, x^2, y^2}$
\pagebreak

$\ideal{x^2, y^2, z}$
\pagebreak

$\ideal{x*y, x^2*y^2*z}$
\pagebreak

$\ideal{x^2, y^2, z^2, t, x*z, y*z}$
\pagebreak

$\ideal{x*y, z}$
\pagebreak

$\ideal{x*y, x*z}$
\pagebreak

$\ideal{x*y}$
\pagebreak

$\ideal{z}$
\pagebreak

$\ideal{1}$
\pagebreak

$\ideal{0}$
\pagebreak

$\ideal{x^2, y^2,
x*z, y*z}$
\pagebreak

$\ideal{x^2, y^2, x*z,
y*z}$
\pagebreak

$x^(a,b,c,d)$
\pagebreak

$x^(a)$
\pagebreak

$x^a$
\pagebreak

$x^b$
\pagebreak

$m$
\pagebreak

$x^a|m|x^b$
\pagebreak

$(a,b)$
\pagebreak

$d(a,b)$
\pagebreak

$x^v$
\pagebreak

$a\leq v\leq
b$
\pagebreak

$(I,S,q)$
\pagebreak

$q$
\pagebreak

$q\textrm{lcm}(\min(I)):x_1\cdots x_n$
\pagebreak

$d_i(t)$
\pagebreak

$x_i^t$
\pagebreak

$\max$
\pagebreak

$(a, b^\prime)$
\pagebreak

$b^\prime$
\pagebreak

$b$
\pagebreak

$b_i^\prime:= a_i+t-1$
\pagebreak

$(a^\prime,b)$
\pagebreak

$a^\prime$
\pagebreak

$a$
\pagebreak

$a^\prime_i := a_i + t$
\pagebreak

$a_i<a_i+t\leq b_i$
\pagebreak

$a_i+t$
\pagebreak

$a_i=b_i$
\pagebreak

$i$
\pagebreak

$a_i<b_i$
\pagebreak

$d(a,b^\prime)\leq
\max$
\pagebreak

$B:=b_i$
\pagebreak

$b_i$
\pagebreak

$B := b_i-1$
\pagebreak

\[  d(a,b^\prime) = d(a,b) - d_i(B) + d_i(a_i+t-1),
\]
\pagebreak

$d(a,b^\prime)\leq \max$
\pagebreak

\[  d_i(a_i+t-1) \leq \max - d(a,b) + d_i(B) =: C.
\]
\pagebreak

$t^\prime:=a_i+t-1$
\pagebreak

$t^\prime$
\pagebreak

$a_i\leq t^\prime<B$
\pagebreak

$d_i(t^\prime)\leq C$
\pagebreak

$t=b_i$
\pagebreak

$t=b_i-a_i$
\pagebreak

\[  d(a,b^\prime)=d(a,b)-d_i(b_i)+d_i(a_i)=:C,
\]
\pagebreak

$d_i(b_i)=d_i(0)$
\pagebreak

$C\leq\max$
\pagebreak

$x_i$
\pagebreak

$d(a^\prime,b)\leq \max$
\pagebreak

\[  d(a^\prime,b) = d(a,b) - d_i(a_i) + d_i(a_i+t),
\]
\pagebreak

\[  d_i(a_i+t) \leq \max - d(a,b) + d_i(a_i) =: C.
\]
\pagebreak

$t^\prime:=a_i+t$
\pagebreak

$a_i< t^\prime\leq b_i^\prime$
\pagebreak

\[  d(a^\prime,b)=d(a,b)-d_i(b_i-1)+d_i(b_i):=C,
\]
\pagebreak

$x_1*x_2$
\pagebreak

$x_2*x_3$
\pagebreak

$x_(a,b)*x_(c,d)$
\pagebreak

$(c,d)$
\pagebreak

\[\left\langle
  \left.
    \left(
      \prod_{i\in F}x_i
    \right)^{n-|F|+1}
  \right|
  \emptyset\neq F\subseteq
  \left\{
    1,\ldots,n
  \right\}
\right\rangle
,\]
\pagebreak

$n$
\pagebreak

$I$
\pagebreak

$S$
\pagebreak

\[  \con I S q = \con {I:p} {S:p} {qp} \cup \con I {S+p} q
\]
\pagebreak

$p$
\pagebreak

$\lcm(min(I))$
\pagebreak

$\min(I)$
\pagebreak

$(\ideal 0, \ideal 0, 1)$
\pagebreak

$(\codeVar{ideal}, \codeVar{subtract},
\codeVar{multiply})$
\pagebreak

$(I:p,S:p,qp)$
\pagebreak

$I:p$
\pagebreak

$S:p$
\pagebreak

$(I,S+\ideal
p,q)$
\pagebreak

\[  \frac{1}{x_i}\gcd(\min(I)\cap\ideal{x_i})
\]
\pagebreak

$\textrm{v} * \textrm{grading}$
\pagebreak

$x ^
 \textrm{v}$
\pagebreak

$a * gcd(a, x_1...x_n)$
\pagebreak

$b\neq 1$
\pagebreak

$u_i<b_i$
\pagebreak

$v_i=0$
\pagebreak

$a=x^u$
\pagebreak

$b=x^v\neq x^0$
\pagebreak

$1=x^{(0,\ldots,0)}$
\pagebreak

$v_i\leq 1$
\pagebreak

$a=x^v$
\pagebreak

$a_i=b_i>0$
\pagebreak

$x_i|a\Leftrightarrow x_i|b$
\pagebreak

$a : b$
\pagebreak

$lcm(a, b)
/ b$
\pagebreak

$x_1>\cdots>x_n$
\pagebreak

$x^(0,0) < x^(0,1) <
x^(1,0)$
\pagebreak

$x_1<\cdots<x_n$
\pagebreak

$x^(0,0) < x^(1,0) < x^(0,1)$
\pagebreak

\end{document}
