\magnification = \magstep1
\noindent
Plan for PURE Math 2012 Seminar
\bigskip
\noindent
{\it Week 1}
\bigskip
\item{} Monday: Polynomials, affine space, affine varieties, parametrizations (Chapter 1, \S\S 1,2,3)
\item{} Tuesday:  Ideals (Chapter 1, \S 4)
\item{} Wednesday:  Polynomials in one variable, division and Euclidean algorithms (Chapter
1, \S 5 -- motivate by determination of ${\bf I}(C)$ for twisted cubic from \S 4)
\item{} Thursday:  Monomial orders in $k[x_1,\ldots,x_n]$ (Chapter 2, \S 2) 
\vfill\eject
\centerline{PURE Math 2012 -- Seminar}
\centerline{Week 1 Discussions -- Polynomials, Varieties, and Algorithms}
\bigskip
\bigskip
\noindent
{\it Note: } Solutions for each day's problems will be due
on the next scheduled ``class day''.  Only one write-up per 
group is required. 
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\bigskip
\noindent
{\it Day 1:  Getting started with varieties}
\bigskip
\noindent
{\it Goals}
\bigskip
Today, we want to start off the discussion part of the seminar
by thinking about the definition of a variety presented earlier
in the morning.

Since this is also the first of the discussion meetings of our
seminar, a few words about this way of working are probably in order.
In these discussion meetings, we will be aiming
for {\it collaborative learning\/} -- that is, for an
integrated group effort in analyzing and attacking the discussion 
questions.  The ideal is for everyone in each of the groups to be fully
involved in the process.  The idea is that, by actively 
participating through talking about the ideas
yourself in your own words, you can come to a better first 
understanding of what is going on than if you simply listen 
to someone else talk about it.  These meetings will also help
prepare you for your work in a research group.

However, it must be said that to get the
most out of this kind of work, you may have to adjust some
of your preconceptions.  In particular:
\medskip
\item{$\bullet$}  This is {\it not a competition\/} in any 
sense.  You and your fellow group members are working as a
team, and the goal is to have everyone understand what the 
group does fully.
\item{$\bullet$}  At different times, it is inevitable that
different people within the group will have a more 
complete grasp of what you are working on and
others will have a less complete grasp.  Dealing with
this a group setting is excellent preparation for real
work in a team; it also offers opportunities for significant
educational experiences:  
\itemitem{a)} If you feel totally ``clueless'' at some point, 
you need to feel free to ask questions
and even pester your fellow group members until the point
has been worked out to your full satisfaction.  (Don't forget,
the others may be jumping to unwarranted conclusions, and
your questions may save the group from pursuing an 
erroneous train of thought!)
\itemitem{b)} On the other hand, when you think you do see 
something, you may need to explain it carefully to others.
(Don't forget, the {\it absolutely best\/} way to make
sure you really understand something is to try to 
explain it to someone else(!)  If you are skipping over an
important point in your thinking, it can become very apparent
when you set out to explain your ideas to a team member.)   
\item{} In short, everyone has important things to contribute, and 
everyone will contribute in different ways at different times.
\vfill\eject
\noindent
{\it Background}
\bigskip
Recall that given a collection of polynomials 
$$f_1(x_1,\ldots,x_n), \ldots, f_s(x_1,\ldots,x_n)$$
the {\it variety\/} ${\bf V}(f_1,\ldots, f_s)$ is just the set of all
points in $k^n$ where {\it all\/} of the polynomials $f_i$ are 
zero {\it simultaneously}.  In symbols:
$${\bf V}(f_1,\ldots, f_s) = \{(a_1,\ldots,a_n) \in k^n : f_i(a_1,\ldots,a_n) = 
{\rm\ for\ all\ } 1 \le i \le s\}.$$
\bigskip
\noindent
{\it Discussion Questions}
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\noindent
A)  Sketch (or describe in words if necessary!) the following varieties:
\medskip
\item{1)}  ${\bf V}(y - x^3 + 3x + 1) \subset {\bf R}^2$, and also in 
${\bf R}^3$, with coordinates $x,y,z$.  
\item{2)}  ${\bf V}(y^2 - x^6) \subset {\bf R}^2$  (what does it mean when
the defining equation factors as this one does:  $y^2 - x^6 = (y - x^3)(y + x^3)$?)
\item{3)}  ${\bf V}(x^2/4 + y^2/9 - 1, xy) \subset {\bf R}^2$
\item{4)}  ${\bf V}(x^2 + y^2 - z^2 - 4, z - 3) \subset {\bf R}^3$
\item{5)}  ${\bf V}(x^2 + y^2 + z^2 - 9, x^2 + y^2 - 4, y - z) \subset {\bf R}^3$ 
\item{6)}  Consider the variety 
$$C = {\bf V}(y^2 - x(x-1)(x-2)) \subset {\bf R}^2$$ 
($C$ is a curve in ${\bf R}^2$ called an {\it elliptic curve\/} --
it is not an ellipse itself, but related curves are involved in
questions like computing arc lengths of parts of ellipses.)  
For which $x$ is it possible to solve the equation of 
the variety for $y \in {\bf R}$? How many $y$-values are there for each 
of these $x$'s?  This curve is {\it symmetric\/} under reflection across
a line -- which line?  Generate an accurate sketch of this curve.  
\bigskip
\noindent
B)  The geometry of varieties has many interesting ``real-world''
applications.  Here is a first taste of one of them.
A robot arm in ${\bf R}^2$ consists of three rigid, straight segments of 
lengths $3,2$, and $1$.  One end of the segment of length 3 is anchored
at the origin, and the other is attached to the segment of length 2, 
which is attached at its other end to the segment of length 1.  
The ``hand'' is at the far end of the segment of length 1.  The 
attachment of the arm at the origin and the joints
between the three segments allow rotations through any angles.
\bigskip
\item{1)}  Draw a picture of the robot arm in a ``typical position''.
\item{2)}  How many variables do you need to specify the ``state''
of the arm, and what are the equations of the variety of possible 
states?
\item{3)}  Describe the set of possible positions of the hand. 
For each of these positions, is there just one configuration of the arm
that places the hand there or more than one?  
\bigskip
\noindent
C)  We can also ask whether a set $S \subset k^n$ 
{\it is a variety}.  That is, we can ask whether $S = {\bf V}(f_1,\ldots,f_s)$
for polynomials $f_i \in k[x_1,\ldots,x_n]$ as above.
\bigskip
\item{1)} Consider $S = \{(x,y) \in {\bf R}^2 : y = \cos(x)\}$
(the graph of the cosine function).  Suppose that $f(x,y)$ is a polynomial
in ${\bf R}[x,y]$ that is zero at every point of $S$.  If 
you substitute a constant value $y_0$ in the range $-1 \le y_0 \le 1$
for $y$, how many roots does the resulting equation $f(x,y_0) = 0$
have?  What does this say about $f$?  (Be careful!  Think
about writing $f$ as a polynomial in $x$ whose coefficients are
polynomials in $y$:
$$f(x,y) = a_n(y)x^n + \cdots + a_1(y)x + a_0(y)$$
Is $S$ a variety?  
\item{2)}  Let $M$ be a positive integer and let 
$S \subset {\bf Z}^2$ be the set of points 
with integer coordinates such that $1 \le x, y \le M$.  
Is $S$ a variety?  Why or why not?
\item{3)} Generalizing part 2, show that
{\it every finite set $S$ in ${\bf R}^2$ is a variety}.  (Hint: one method
begins by showing that there is always a rotated $(x',y')$
coordinate system such that the $x'$-coordinates of the points in $S$ 
are {\it distinct}.)
\bigskip
\noindent
D)  Problem 13 of ``IVA'', Chapter 1, \S 3.  
(Note: Try to work out the implicit equation both ways from the 
Hint in the problem to check your work.  If you don't recall the vector cross product 
and the way it can be used to get a normal vector for a plane, we will 
show that to you.)
\bigskip
\noindent
%{\bf Day 2: Parametrizations of Varieties}
%\bigskip
%\noindent
%{\it Background}
%\bigskip
%A portion of our work this summer will involve the relation between
%the following two ways to represent geometric objects:
%\bigskip
%\item{a)}  as the {\it sets of solutions of one or more ``implicit''
%equations} 
%\item{b)}  as the {\it images of parametrizations\/}.
%\bigskip
%\noindent
%The standard description of a variety in affine space $k^n$
%$${\bf V}(f_1,\ldots,f_s) = \{(a_1,\ldots,a_n) : f_i(a_1,\ldots,a_n) = 0,
%1 \le i \le s\}$$
%is an example of the first kind above.  For instance, the circle
%of radius 2 with center at $(1,2)$ is the variety 
%$$V((x-1)^2 + (y-2)^2 - 4)\leqno(1)$$
%given by the equation $(x-1)^2 + (y-2)^2 - 4 = 0$.  But there is also another
%way to describe the same circle -- as the image of a parametrization
%mapping such as 
%$$\alpha(t) = (1 + 2\cos(t),2 + 2\sin(t)).\leqno(2)$$
%We get the full circle, traced out once, for $0 \le t \le 2\pi$.  
%
%Each of these ways to describe objects is useful.  If we want to 
%program a computer to draw the object, for instance, then a 
%{\it parametrization\/} is extremely good to have.  With the parametrization
%(2) in hand, for instance, we could ``plug in'' a collection of (closely-spaced)
%$t$ values, then ``connect the dots'' to draw the curve.  (This is exactly 
%what programs like Mathematica do for parametric plotting of curves!)  However, if
%we want to determine whether a given point is in the geometric locus
%(or within a given distance, etc.),
%then {\it implicit equations} are much better, since we only need
%to substitute the coordinates of the point in question into the
%right equations to determine the answer to our question.
%
%\bigskip
%\noindent
%{\it Discussion Questions}
%\bigskip
%\item{A)}  Problem 5 of ``IVA'', Chapter 1, \S 3.  The hyperbolic
%functions are defined as follows:
%$$\cosh t = {e^t + e^{-t}\over 2}, \qquad \sinh t = {e^t - e^{-t} \over 2}$$ 
%\item{B)}  Problem 8 of ``IVA'', Chapter 1, \S 3. 
%\item{C)}  Problem 13 of ``IVA'', Chapter 1, \S 3.  
%(Note: Try to work out the implicit equation both ways from the 
%Hint to check your work.  If you don't recall the vector cross product 
%and the way it can be used to get a normal vector for a plane, we will 
%show that to you.)
%\item{D)}  Problems 14, 15 from ``IVA'', Chapter 1, \S 3 (see the discussion
%of B\'ezier cubics in the text).  
%\bigskip
%\noindent
%{\it Comment:}  In computer-aided geometric design (CAGD), collections
%of several Bezier cubics are {\it linked together}
%to form more complex curves or shapes.  You will see an example of this
%in today's lab meeting.  
\bigskip
\noindent
{\it Day 2:  Ideals}
\bigskip
\noindent
{\it Background}
\bigskip
As we saw in lecture earlier, the ``defining equations'' $f_i = 0$
of a variety given as 
$V = {\bf V}(f_1,\ldots,f_s)$ are never {\it the only\/}
polynomial equations that are satisfied at all points of the 
variety.  We can always look at other polynomials of the form
$$f = g_1 f_1 + \cdots + g_s f_s\leqno(3)$$
(where the $g_i$ are any polynomials at all in the same set of 
variables), and we get other polynomials that vanish at all points
of $V$.  The polynomials $f$ in (3) are the elements of the 
{\it ideal}
$$I = \langle f_1,\ldots, f_s\rangle \subset k[x_1,\ldots,x_n].$$
An ideals is a non-empty collection $I$ of polynomials that is 
\medskip
\item{$\bullet$} closed under sums, and 
\item{$\bullet$} closed under multiplication by arbitrary polynomials
$g \in k[x_1,\ldots,x_n]$.  
\medskip
\noindent
Given a variety $V$, we can also produce an ideal by considering the 
collection of {\it all polynomials that vanish at every point of $V$}:
$${\bf I}(V) = \{f \in k[x_1,\ldots,x_n] : f(a_1,\ldots,a_n) = 0,
{\rm\ all\ } (a_1,\ldots, a_n) \in V\}.$$
\vfill\eject
\noindent
{\it Discussion Questions}
\bigskip
\noindent
A)  In this problem, we will develop a way to tell whether two
ideals are equal.
\bigskip
\item{1)}  Let's start with an example.  
We ask: Are $I = \langle 2x^2 + 3y^2 - 11, x^2 - y^2 - 3\rangle$
and $J = \langle x^2 - 4, y^2 - 1\rangle$ equal or not?  Show
that both $2x^2 + 3y^2 - 11$ and $x^2 - y^2 - 3$ are in $J$
by finding polynomials $A_1,B_1$ and $A_2,B_2$ such that
$$\eqalign{2x^2 + 3y^2 - 11 &= A_1 (x^2 - 4) + B_1 (y^2 - 1)\cr
x^2 - y^2 - 3 &= A_2 (x^2 - 4) + B_2 (y^2 - 1)\cr}$$
Explain why this shows $I \subseteq J$.
Similarly, decide whether both $x^2 - 4$ and $y^2 - 1$
are in $I$ by seeing whether there exist polynomials $C_1,D_1$
and $C_2,D_2$ such that 
$$\eqalign{x^2 - 4 &= C_1 (2x^2 + 3y^2 - 11) + D_1 (x^2 - y^2 - 3)\cr
y^2 - 1 &=  C_2 (2x^2 + 3y^2 - 11) + D_2 (x^2 - y^2 - 3)\cr}$$
Are $I$ and $J$ equal?  Why or why not?
\item{2)}  Now, consider this question
in general.  Let $I$ be an ideal, and $g_1,\ldots,g_t$ be a 
collection of polynomials.  Show that $\langle g_1,\ldots,g_t \rangle
\subseteq I$ if and only if $g_i \in I$ for all $i = 1, \ldots, t$.
\item{3)}  Use part 1 to develop a general ``test'' to perform to 
determine whether two ideals $I= \langle f_1,\ldots, f_s\rangle$
and $J = \langle g_1,\ldots, g_t\rangle$ are equal.
\bigskip
\noindent
B)  Let $C$ be the curve in ${\bf R}^3$ parametrized by 
$$\alpha(t) = (t,t^3,t^5)$$
\bigskip
\item{1)}  Show that $f_1 = y - x^3$ and $f_2 = z - x^5$
are elements of ${\bf I}(C)$.  
\item{2)}  Prove that $C$ is a variety.
\item{3)}  Following what the text does for the twisted cubic, determine
${\bf I}(C)$.
\bigskip
\noindent
C)  Let $I$ be an ideal in $k[x_1,\ldots,x_n]$.  Let
$\sqrt{I}$ be the collection of all polynomials $f$ such that some 
power of $f$ is in $I$  (the smallest power that ``works'' can be different
for different $f$).  $\sqrt{I}$ is called the {\it radical} of $I$, since it consists
of all polynomials that are ``$\ell$-th roots'' of elements of $I$, 
for all $\ell \ge 1$.  In symbols,
$$\sqrt{I} = \{ f \in k[x_1,\ldots,x_n] : f^\ell \in I, {\rm\ for\ some\ } 
\ell \ge 1\}$$
\item{1)}  For example, explain why $f = x + y$ is in 
$\sqrt{I}$ for $I = \langle x^4 + y^4, xy\rangle$.
\item{2)}  Show that $\sqrt{I}$ is closed under products
by all $g \in k[x_1,\ldots,x_n]$.
\item{3)}  Show that $\sqrt{I}$ is closed under sums.  (That is,
if $f \in \sqrt{I}$ and $g\in \sqrt{I}$ so $f^\ell \in I$, and 
$g^m \in I$ for integers $\ell, m \ge 1$ (not necessarily the same), 
you need to show there is some $p\ge 1$ such that $(f+g)^p \in I$.  
What $p$ work?) 
\item{4)}  What do parts 2 and 3 say about $\sqrt{I}$?
\bigskip
\noindent
D)
\item{1)}  Show that ${\bf V}(I) = {\bf V}(\sqrt{I})$ for all ideals $I$.
\item{2)} An ideal $I$ is said to be a {\it radical ideal\/} if $\sqrt{I} = I$.
Show that if $V$ is variety, then ${\bf I}(V)$ is a radical ideal.  
\bigskip
\noindent
{\it Day 3:  Algorithms}
\bigskip
\noindent
{\it Background}
\bigskip
In class today, we studied our first examples of {\it algorithms},
or step-by-step procedures for performing a particular computation.  
The two examples we looked at were:
\bigskip
\item{$\bullet$} the {\it division algorithm\/} in $k[x]$,
\item{$\bullet$} the {\it Euclidean algorithm\/} for gcd's in $k[x]$.
\bigskip
\noindent
Today, we want to work through a few examples, following the ``pseudo-code''
description of the algorithm (as if you were a computer, yourself, following
a program!) Then we will want to develop an extension of the
Euclidean algorithm that, together with the gcd, also produces polynomials 
$A$ and $B$ satisfying
$$gcd(f,g) = Af + Bg.$$
\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\noindent
A)  Work through the division algorithm in ${\bf Q}[x]$ step
by step (see page 38 in ``IVA''), dividing
$g(x) = 2x^3 + 3x - 5$ into $f(x) = x^5 - x^4 + x^2 - 2x + 1$.  At the 
start of each pass through the main loop, what are the values of 
$q,r$?  Show that the equation $f = qg + r$ is always true.  (This
is an example of a ``loop invariant'' -- using these is often important
in developing an algorithm or proving that an algorithm is correct.)
\bigskip
\noindent
B)  Work through the Euclidean algorithm step by step (see page 41)
to compute the gcd of $f$ and $g$ from part A.  What are the values
of $h$,$s$ at the start of each pass through the loop?
\bigskip
\noindent
C)  Do Problem 10 in Chapter 1, \S 5 of ``IVA''.  (Hint: Read the 
given Hint carefully and think about the idea of ``loop invariants''
from above.)  ``Run'' your
algorithm on the example from questions A and B, and check your 
results.  
\bigskip
\noindent
{\it Comment:}
Although this ``extended Euclidean algorithm'' is also available
as a ``built-in'' command in Sage and other computer algebra systems, 
you will also see how to code it as a Sage procedure in a lab meeting.  
\vfill\eject
\noindent
{\it Day 4: Monomial Orders}
\bigskip
In class today, we introduced the definition of a {\it monomial order\/} in 
the polyomial ring $k[x_1,\ldots,x_n]$.  This was an ordering $>$ on the 
monomials $x^\alpha$ (or equivalently on the exponent vectors $\alpha$) such 
that:
\bigskip
\item{1)}  $>$ is a total (linear) order on vectors with nonnegative integer 
components. (This means that for every pair of monomials $x^\alpha$ and 
$x^\beta$, exactly one of the following is true:  either $x^\alpha > x^\beta$,
or $x^\alpha = x^\beta$, or $x^\beta > x^\alpha$.)
\item{2)}  For all $\alpha$, $\beta$, $\gamma$, we have
$$\alpha > \beta \Rightarrow \alpha + \gamma > \beta + \gamma$$
\item{3)}  $>$ is a well-ordering.  (In other words, every non-empty subset of 
monomials or exponent vectors has a smallest element under $>$).
\bigskip
\noindent
Today, we want to think about some of the implications of this definition.
\bigskip 
\noindent
{\it Discussion Questions}
\bigskip
\noindent
A)  To practice with monomial orders, do Problems 1,2 from Chapter 2, 
\S 2 of ``IVA''.
\bigskip
\noindent
B)  Show that in $k[x,y]$, the graded lexicographic and graded reverse 
lexicographic orders are the same.  Is this true for $n > 2$?
\bigskip
\noindent
C)  Suppose we try to define an order $>$  by omitting the first part of 
the definition of  the graded lexicographic order, so that $\alpha > \beta$ if 
the rightmost entry in $\alpha - \beta$
is negative.  Do we get a monomial order?  Why or why not?
\bigskip
\noindent
D) We could also try to define an order $>_w$ by comparing the ``total weights'' 
of two monomials with respect to a given weight vector $w$ on the variables.  
That is, for a monomial $x^\alpha$, we think of the variable $x_i$ as having 
weight $w_i$ (from the vector $w$).  We compute
$\alpha\cdot w = \alpha_1 w_1 + \cdots + \alpha_n w_n$ and say 
$x^\alpha >_w x^\beta$ if $\alpha \cdot w > \beta\cdot w$.
\bigskip
\item{1)}  Let $n = 2$ and $w = (3,5)$.  Is $>_w$ a monomial order?  
Why or why not?
\item{2)}  Let $n = 2$ again and $w = (1,\sqrt{2})$.  Same question.
\item{3)}
What property of the components of the vector $w \in {\bf R}^n$ would imply that $>_w$ 
{\it does} define a monomial order on $k[x_1,\ldots,x_n]$?   (The fact that 
$\sqrt{2}$ is not a rational number in part 2 is an important clue, but it is not 
the whole story!)
\bye



