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\centerline{\textbf{Dartmouth College}} \centerline{Mathematics 81/111
  --- Homework 1}


\begin{center}
  \textbf{Some basic definitions concerning algebraic sets}
\end{center}


Let $k$ be a field, and let $k[x_1, \dots, x_n]$ the polynomial ring
in $n$ variables with coefficients in $k$. For
$f \in k[x_1, \dots, x_n]$ and $P = (a_1, \dots, a_n) \in k^n$, we
write $f(P)$ for $f(a_1, \dots, a_n)$, and define the \textbf{zero
  set} of $f$ by
\[
  Z(f) = \{P \in k^n \mid f(P) = 0\}.
\]
For example, consider three real plane curves $Z(f)$, i.e., where
$f \in \R[x,y]$.
 
\vspace{-.2in}
\begin{figure}[h]
  \centering
  \subfloat[$Z(y^2 -
  x(x^2-1))$]{\label{fig:nonsingular}\includegraphics[width=0.3\textwidth]{images/EC}}
  \subfloat[$Z(y^2-x^2(x+1))$]{\label{fig:node}\includegraphics[width=0.3\textwidth]{images/singular}}
  \subfloat[$Z(y^2-xy -x^2y +
  x^3)$]{\label{fig:reducible}\includegraphics[width=0.3\textwidth]{images/reducible}}
  % \caption{Affine Plane Curves}
  % \label{fig:algebraic_curves}
\end{figure}

\begin{enumerate}
\item For a subset $S \subseteq k[x_1, \dots, x_n]$, define the
  \textbf{zero set of $S$} to be the common zeros of all the elements
  of $S$, that is
  \[Z(S) = \{P \in k^n \mid f(P) = 0 \textrm{ for all } f \in S\} =
    \bigcap_{f\in S} Z(f).
  \]
\item A subset $X \subseteq k^n$ is called an (affine)
  \textbf{algebraic set} if $X = Z(S)$ for some
  \mbox{$S \subseteq k[x_1, \dots, x_n]$.}  Let $\A$ denote the set of
  all affine algebraic subsets of $k^n$. It is trivial to check that
  $Z(S) = Z(\,( S)\,)$, where $( S)$ is the ideal generated by the set
  $S$ in $k[x_1, \dots, x_n]$.

\item Let $\I$ denote the set of ideals in $k[x_1, \dots, x_n]$, so
  every element of $\A$ is of the form $Z(J)$ for some $J \in \I$.

\item For any subset $Y \subseteq k^n$ define the \textbf{ideal of
    $Y$}, $I(Y) \in \I$, by
  \[I(Y) = \{f \in k[x_1, \dots, x_n] \mid f(P) = 0 \textrm{ for all }
    P \in Y\}.
  \]
\end{enumerate}



We have now defined two functions: $Z:\I \to \A$ which maps subsets of
$k[x_1, \dots, x_n]$ to affine algebraic sets, and a function
$I:\A \to \I$, which maps subsets of $k^n$ to ideals.


\newpage

\begin{center}
  \textbf{The Exercises}
\end{center}
In the first few exercises, we establish some basic properties of the
maps defined above.

\begin{questions}

  \question Show that the union of two algebraic sets is an algebraic
  set.  Show that the intersection of any family of algebraic sets is
  an algebraic set.  Show that the empty set and the whole space
  ($k^n$) are algebraic sets.

  If we take the algebraic sets as the closed sets in $k^n$, the
  properties above for closed sets defines a topology on $k^n$, called
  the \textit{Zariski topology}.




  \question We shall take as given that
  $I(\emptyset) = k[x_1, \dots, x_n]$.  Show that $I(k^n) = \{0\}$ if
  and only if $k$ is an infinite field.  Also show that if
  $P = (a_1, \dots, a_n) \in k^n$, then\newline
  $I(P) = ( x_1-a_1, x_2-a_2, \dots, x_n - a_n)$, the ideal generated
  by $\{x_1-a_1, x_2-a_2, \dots, x_n-a_n\}$.
 

  \question Some basic correspondences:
  \begin{parts}
    \part Show that $I(Z(S)) \supseteq S$ for any set
    $S \subseteq k[x_1,\dots, x_n]$, and $Z(I(X)) \supseteq X$ for any
    set $X \subseteq k^n$.  \textbf{Hint:} First establish the very
    useful observation that both functions $I$ and $Z$ are inclusion
    reversing.

 

    \part Deduce that $Z(I(Z(S))) = Z(S)$ for any set
    $S \subseteq k[x_1,\dots, x_n]$, and $I(Z(I(X))) = I(X)$ for any
    set $X \subseteq k^n$, that is $Z(I(V)) = V$ for any algebraic set
    $V$, and $I(Z(J)) = J$ for any $J$ which is the ideal of a set in
    $k^n$.

 


    \part Show that for any set $Y \subseteq k^n$, we have
    $Z(I(Y)) = \overline Y$, the closure of $Y$ in the Zariski
    topology.
  \end{parts}

 
  \begin{nnremark}
    If $k$ is algebraically closed (e.g., $k = \C$), then for any
    ideal $J \subseteq k[x_1, \dots, x_n]$, Hilbert's Nullstellensatz
    shows that $I(Z(J)) = \sqrt J$ where
    \[\sqrt J = \{ f \in k[x_1, \dots, x_n] \mid f^r \in J \textrm{
        for some } r \in \Z_+\}, \textrm{ is called the
        \textbf{radical} of $J$.}
    \]


  \end{nnremark}

  \question
  \begin{minipage}[t]{\textwidth}
    Consider the four-leaved rose given as the polar plot of
    $r = \sin(2\theta)$.  We wish to show that this curve is an
    algebraic set.
   
    \begin{wrapfigure}{l}{0.4\textwidth}%
      \vspace{-10pt}
      \fbox{\includegraphics[width=0.4\textwidth]{images/rose}}
    \end{wrapfigure}
    
    \mbox{\hfill}\newline \mbox{\hfill}\newline Start with the usual
    change of coordinates from polar to Cartesian coordinates:
    \[x = r\cos \theta;\qquad y = r\sin\theta;\qquad(x^2 + y^2 =
      r^2).\] First show that every point of the polar curve is on the
    algebraic set $Z((x^2 + y^2)^3 - 4x^2y^2));$ a simple trig
    identity may be useful. Then show the converse.  The converse
    requires a bit of thought (and explanation).
  \end{minipage}
  % \vspace{.95in}
  \newpage

  \question Let $k$ be an infinite field, $Y$ be the algebraic set
  $Z(y-x) \subset k^2$, and let $X =Y \setminus \{(1,1)\}$, that is,
  the line minus the point $(1,1)$.  We want to show that $X$ is not
  an algebraic set.  Proceed by contradiction, assuming $X = Z(S)$ for
  some set of polynomials.  Since $Z(S) = Z(\la S\ra)$, we may assume
  (by the Hilbert Basis Theorem (soon to come)) that
  $S = \{f_1, \dots, f_n\}$ is a finite set.  Here each
  $f_i\in k[x,y]$.
  \begin{parts}
    \part Show that for any $f\in k[x,y]$, $f \in I(X)$ implies
    $f \in I(Y)$.
    \part Use previous problems in this homework set to obtain a
    contradiction.
  \end{parts}

  \renewcommand{\P}{{\mathbf P}} \newcommand{\Q}{{\mathbf Q}}
  \question In the last two problems, we consider cubic Berstein
  polynomials, and their associated plane curves, called B\'ezier
  curves.  While Berstein's work was involved with approximation
  theory, they were used in 1959 by Casteljau to help design car
  bodies for Citro\"en.  That use was better publicized (1962) by
  B\'ezier, an automobile designer for Renault, and so now has his
  name attached to them.  They are also fundamental in the PostScript
  language and play a crucial role in Knuth's METAFONT system.

  The idea is quite simple.  You want to produce a curve knowing the
  starting and ending points, as well as the tangent directions at the
  endpoints.  To do this, we need four (control) points:
  $\P_0, \dots, \P_3$ with $\P_j$ having coordinates $(x_j,y_j)$.
  Points on the cubic B\'ezier curve are given parametrically by:
  \[\P(t) = (x(t), y(t)) = (1-t)^3\P_0 + 3t(1-t)^2\P_1 + 3t^2(1-t)\P_2
    + t^3\P_3, \quad 0\le t\le 1.
  \]
  \begin{parts}
    \part Trivially, we see that $\P(0) = \P_0$ and $\P(1) = \P_3$.
    Show that the tangent directions at the endpoints are parallel to
    the vectors from $\P_0$ to $\P_1$, and from $\P_2$ to $\P_3$.
    Thus positioning the control points $\P_1$ and $\P_2$ allows a
    great deal of flexibility in the shape of the curve.
    
    \part Explain what happens if you replace $\P_1$ by a point
    $\P_1'$ which lies on the line through $\P_0$ and $\P_1$, but
    which is nearer or farther away from $\P_0$ than $\P_1$.  As a
    visual cue, see the pictures below which show the effect of moving
    one of the two control points $P_1$ or $P_2$.

    \includegraphics*[width=3in]{images/bezier}
    \includegraphics*[width=3in]{images/bezier2}


  \end{parts}

  \question Next we want to show that a B\'ezier curve always lies
  within the convex hull (control polygon) determined by the four
  control points.
  \begin{parts}
    \part Recall that a subset $C \subseteq \R^2$ is call
    \textit{convex} if for points $\P, \Q\in C$ the line segment
    joining $\P$ to $\Q$ lies in $C$.  Show that if $\P, \Q\in C$,
    then all points $t\P + (1-t)\Q \in C$, for $0\le t\le 1.$
    \part Show that if $\P_1, \dots, \P_n$ lie in a convex subset $C$,
    then so does $\displaystyle \sum_{i=1}^n t_i\P_i$ where the $t_i$
    are nonnegative real numbers whose sum is 1.

    \part Show that every point on the B\'ezier curve lies in the
    convex polygon determined by the four control points which define
    it.  Some samples are pictured below:

    \includegraphics*[width=3in]{images/bezier_convex}

  \end{parts}
   

\end{questions}



\end{document}






\question An algebraic set $V \subseteq k^n$ is \textbf{reducible}
  if $V = V_1 \cup V_2$ where $V_1, V_2$ are algebraic sets in $k^n$
  and $V_i \ne V$, $i = 1,2$.  Otherwise $V$ is \textbf{irreducible},
  and $V$ is called an \textbf{affine algebraic variety}. Note that
  the third example on the first page is reducible: $Z(y^2-xy -x^2y +
  x^3) = Z((y-x)(y-x^2)) = Z(y-x) \cup Z(y-x^2)$.

  Show that an algebraic set $V$ is irreducible if and only if $I(V)$
  is a prime ideal.


\question \textbf{Inverse limits.}  Let $I$ be a partially ordered
  set (see Appendix A5 if unsure of the definition), and for each $i
  \in I$, let $A_i$ be a ring (also works in other categories).
  Suppose that for each $i,j \in I$, $i \le j$ we are given a
  homomorphism $\varphi_{ji}: A_j \to A_i$ satisfying:
  \begin{parts}
  \part $\varphi_{ji} \circ \varphi_{kj} = \varphi_{ki}$ whenever $i
    \le j \le k$, and
  \part $\varphi_{ii}$ is the identity map for all $i \in I$.
  \end{parts}
  Let $P = \{ (a_i) \in \prod_{i\in I} A_i \mid a_i =
  \varphi_{ji}(a_j) \textrm{ whenever } i \le j\}$; $P$ is denoted $P
  = \varprojlim A_i$, and is called the \textit{inverse} or
  \textit{projective} limit of the $A_i$.

  \begin{parts}
  \part Show that $\varprojlim A_i$ is a ring under the operations
    inherited from the direct product.




  \part (Universal mapping property).  For each $k \in I$, let
    $\varphi_k: \varprojlim A_i \to A_k$ be the standard projection.
    Suppose that $R$ is any ring such that for each $i \in I$ there is
    a homomorphism $\pi_i: R \to A_i$ so that $\pi_i =
    \varphi_{ji}\circ \pi_j$ whenever $i \le j$.  Show that there is a
    unique homomorphism $\pi: R \to \varprojlim A_i$ such that
    $\varphi_i \circ \pi = \pi_i$ for all $i \in I$.



  \end{parts}




  \question \textbf{An example of an inverse limit.} Let $p$ be a
  prime, $I = \Z_+$ with the usual partial ordering, and put $A_i =
  \Z/p^i\Z$.  For $i \le j$, let $\varphi_{ji}:A_j \to A_i$ be the
  natural surjective ring homomorphism (Proposition 4.21).  The
  inverse limit of the $A_i$, $\varprojlim \Z/p^i\Z$, is denoted
  $\Z_p$ and called the ring of $p$-adic integers.
  \begin{parts}
  \part Show that every element of $\Z_p$ can be written uniquely as a
    formal sum $b_0 + b_1 p + b_2 p^2 + b_3 p^3 + \cdots$ where each
    $b_i \in \{0, 1, \dots, p-1\}$.  Represent each element in $A_i =
    \Z/p^i\Z$ as a least residue $\pmod p$ and express the element in
    its base $p$ expansion.  Characterize the action of the maps
    $\varphi_{ji}$ on these representations.  Note that this implies
    that $\Z_p$ is an uncountable set.




  \part Show that an element $\alpha = b_0 + b_1 p + b_2 p^2 + b_3 p^3
    + \cdots \in \Z_p$ is a unit (in the ring) if and only if $b_0 \ne
    0$.

        


  \end{parts}






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