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\begin{document}
\title{Information on Mathematics 421\\
Algebraic Geometry, Fall 1996}
\date{November 15, 1996}
\author{Andrew J. Sommese}
\maketitle
\tableofcontents
\section*{General information}
This course is new this year and therefore the material will be
adjusted somewhat for next year.
The enrollment is currently stable at 20. The maximum enrollment was
23 or 24. All students are mathematics majors and all but one of the
students are seniors. Overall the students are quite reasonable and I
think that they are enjoying the class (as am I).
I give a homework assignment every week. Usually four problems: I have
appended the problems given so far this year. I have broken the class
up into collaborative learning groups of three and four students. Each
group works as a group on the homework, which I grade and return. I
also put answers to the homework on the network. Besides the homework,
which is worth 100 points, I give two tests worth 100 points each and a
final worth 150 points. I have appended a copy of the first test and
my handout with basic information.
I plan to use the computer more in the class next year. I often
suggest using maple for assignments, and I have begun giving
demonstrations in class.
Besides the three classes on Monday, Wednesday, and Friday, I meet my
class every Tuesday from 7:15 PM to 8:15 PM to answer questions, review
and fill in gaps in people's backgrounds, and go over problems on the
homework or on tests that caused difficulties.
We have already covered the real and complex projective line and plane.
We have covered the homogeneous form of the tangent line to a curve,
Euler's theorem on homogeneous functions, and how to find the singular
points of plane curves. We have covered B\'ezout's theorem (with the
proof so far only for two curves one of which is a line or a conic,
though as a consequence of the basic theory of resultants which
developed in class have shown how to find all the solutions of a system
of two polynomials in two unknowns over the complex numbers). We spent
quite some time going over complex numbers, which many students had (to
me shockingly) never seen. I went over calculus in the plane, i.e.,
power series, the Cauchy-Riemann equations, use of Green's theorem in
the plane, i.e., the Cauchy integral theorem. I also have covered
divisors of rational functions and one forms on the line, and covered
calculation of the Euler characteristic of Riemann surfaces. My most
hopeful plan was to study the projective plane and algebraic curves,
leading up to a proof of special cases of Riemann-Roch for curves and
applications of it. Some of this was too ambitious, and I have
moderated my plans. Also a good number of the students have forgotten
a fair amount of the algebra and calculus that they learned. I shifted
to covering very concrete topics, e.g., proof of partial fraction
decomposition; the proof of basic results about resultants and
discriminants (including the proof that polynomial rings over fields
are unique factorization domains) leading up to a constructive proof of
B\'ezout's theorem. This has been going very well. I will then define
affine varieties and their functions using polynomial rings. I will
explain the Nullstellensatz and set up the usual dictionary between
affine varieties and quotient rings of the ring of polynomials. I will
if possible prove the Nullstellensatz. I will then cover the basic
results on computational algebraic geometry, i.e., Groebner bases.
This is very down to earth, and the students like the concrete
algebra. I will try to cover the basic theory of symmetric functions
and some theory of the symmetric group, which fits in nicely with the
above.
The book by Reid turned out to be unsuitable for the class. Besides
being a little too hard, the material in any part was too dependent on
material already covered. I wrote some notes for some topics covered
in the class (they are on the network), but would still like them to
have a book I can assign reading in. I have followed Walker's {\em
Algebraic Curves}, which I put on reserve in the library, for my
presentation of Unique factorization domains, resultants,
discriminants. I have done this carefully and gone over it enough that
it is understood. Next year I plan to use the book {\em Ideals,
Varieties, and Algorithms} by Cox, Little, and O'Shea, which was out of
print (it was between editions) at the the time I had to order a course
book.
\newpage
\section{Syllabus for Next Year}For next year my plan would be to
\begin{enumerate}
\item spend the first few weeks going over complex numbers and enough
calculus in the plane to prove the fundamental theorem of algebra;
\item develop the basic theory about the projective line and plane
over the real and complex numbers and algebraic plane curves,
especially those of degree up to three;
\item introduce tangent lines and singular points;
\item study polynomials and rational functions on the line and plane
(including partial fraction expansions, one and two-forms, linear
fractional transformations, and basic elimination theory);
\item define affine varieties and their functions using polynomial
rings;
\item explain the Nullstellensatz and set up the dictionary between
affine varieties and quotients of the ring of polynomials (and if
possible prove the Nullstellensatz);
\item cover the basic results on computational algebraic geometry,
i.e., Groebner bases;
\item cover the basic theory of symmetric functions and some theory
of the symmetric group.
\end{enumerate}
\subsection{Requirements for the Course} The requirements should be
increased. Besides the calculus sequence Mathematics 125 to 226; and
the courses on algebraic structures and linear algebra; I would like to
add real analysis or ordinary differential equations.
\newpage
\section{Class Handout}
\begin{tabbing}
Text: \=Element \=Undergraduate Algebraic Geometry, by Gilbert and Gilbert\kill
Instructor: \>\> Andrew Sommese\\
\> \> 231 CCMB (On Juniper, just south of the main library)\\
\>\> Phone: 631-6498; e-mail: sommese.1@nd.edu\\
Text: \>{\em Undergraduate Algebraic Geometry}, by M. Reid
\end{tabbing}
%\smallskip
I am in my office almost all of every weekday, and encourage students
to visit any time. If you just come to my office you will probably find
me, but if you set up a time with me before hand, then you can be sure
I will be there.
\paragraph{Examinations, homework and grades}
There will be two departmental examinations and one final examination
(whose dates and times are listed below). Each departmental exam is a
one-hour exam and will be worth 100 points. The final exam is a
two-hour exam and will be worth 150 points. Homework will be worth 100
points. The final exam will cover all the material of the course. The
total number of possible points for the semester is 450. The numerical
break points for letter grades (A, A-, B+,\ldots) will be based only on
the test scores and the homework.
Homework will be assigned regularly, and is an integral part of the
course. I ask students to form groups of three (and one or two
groups of size four depending on the number of students in the class
mod 3) to do all the assignments. If there are people who would
like to be in the same group, please let me know by the end of class on
Friday, August 30. I will hand out the lists of members of the class
by groups on Monday, September 2. If the class enrollment changes I
might have to add members to a few groups. Typically I will give the
week's assignment on a Monday and collect it the following Monday. I
strongly encourage students to see me if there is anything they are
unclear on would like to know more about.
Both examinations and the homework are conducted under the honor code.
People within a group are graded together on the homework assignments,
and are expected to work together. People in different groups are
encouraged to discuss the mathematics, but should not discuss how to do
the week's assignment before it is handed in!
A student who misses an examination will receive zero points for that
exam unless he or she has written permission from the Vice President
for Student Affairs.
\paragraph{Exam Dates} \begin{tabbing} Exam 1:\ \ \ \ \ \=Monday,
February 7, 1994\ \ \ \ \=(in class)\kill Exam 1: \> Wednesday,
October 2, 1996 \> (in class)\\ Exam 2: \> Friday, November 15, 1996 \>
(in class)\\ Final: \> Thursday, December 19, 1996, 8--10AM\\ \>
(Location of final exam will be announced later.) \end{tabbing}
\newpage
\section{Homework}Generally there has been one week give for the
completion of each assignment.
\hspace*{-0.5in} {\bf Problems due September 9}
\begin{problem}[2 points]For an inscribed $n$-sided regular polygon in
the unit circle, show that the the product of the lengths of the $n-1$
segments from one fixed vertex to the remaining vertices equals $n$ for
all $n\ge 2$.
\end{problem}
\begin{problem}[5 points]Find the singular points of the curve
$zxy-z^3=0$. Write the equation and sketch the graph in the coordinates
for the standard Euclidean $\reals^2$ centered at $[0,0,1]$ and also at
$[1,0,0]$.
\end{problem}
\begin{problem}[4 points]Show that the Klein quartic,
$x^3y+y^3z+z^3x=0$, is smooth. Show also that it is invariant under
the projectivity $[x,y,z]\to[y,z,x]$ and also under the projectivity
$[x,y,z]\to [x,\alpha^2 y,\alpha^3 z]$ where $\alpha$ is a $7$-th
root of unity, i.e., $\alpha^7=1$.
\end{problem}
\hspace*{-0.5in} {\bf Problems due September 18}
\begin{problem}[3 points]Find all automorphisms of $\pn 1_\comp$, $f:
[z,w]\to [az+bw,cz+dw]$, such that $f(f([z,w]))=[z,w]$. Find all
automorphisms of $\pn 1_\comp$, $f: [z,w]\to [az+bw,cz+dw]$ such that,
in addition to $f(f([z,w]))=[z,w]$, $f([1,1])=[1,1]$ and
$f([2,1])=[2,1]$. (Find all means in particular to show the
automorphisms that you find are all of the automorphisms.)
\end{problem}
\begin{problem}[2 points]Find all automorphisms of $\pn 1_\comp$, $f:
[z,w]\to [az+bw,cz+dw]$, such that $f([0,1])=[0,1]$, $f([1,1])=[1,1]$,
and $f([1,0])=[1,0]$. (Find all means in particular to show the
automorphisms that you find are all of the automorphisms.)
\end{problem}
\begin{problem}[4 points]Write down the homogeneous degree one
polynomial that defines the tangent line in $\pn 2_\comp$ to the curve
$C$ defined by $x^{d-1}y+y^{d-1}z+z^{d-1}x=0$ at $[0,0,1]$. What
points does this line meet $C$? What are their multiplicities.
\end{problem}
\hspace*{-0.5in}
{\bf Problems due September 25}
\begin{problem}[3 points] We have a natural group homomorphism from
$\SL 2{\comp}$, the $2\times 2$ complex matrices
$\displaystyle\twoMatrix a b c d$ with determinant $1$, to the group
$\PL 1{\comp}$ of automorphisms, $[z,w]\to [az+bw,cz+dw]$, of $\pn
1_{\comp}$. What is the kernel of this mapping? Show that the mapping
$\SL 2{\comp}\to \PL 1{\comp}$ is onto.
\end{problem}
\begin{problem}[3 points] Find the points in $\pn2_{\comp}$ on the
curves $x-y=0$ and $x^{10}+y^{10}-2z^{10}=0$. What are their
multiplicities? Explain why this is or isn't compatible with
B\'ezout's theorem.\end{problem}
\begin{problem}[5 points] Find the points in the complex $\pn2_{\comp}$
on the curves $z^2-xy=0$ and $x^{10}+y^{10}-2z^{10}=0$. What are their
multiplicities? Explain why this is or isn't compatible with
B\'ezout's theorem. {\small HINT: Use the parameterization of
$z^2-xy=0$ given by $[a,b]\to [a^2,b^2,ab]$.} \end{problem}
\hspace*{-0.5in} {\bf Problems due October 2}
\begin{problem}[3 points] Compute $d\omega$ where
$\omega=x^2dx+y^4dy$. Let $\displaystyle D= \{(x,y) \in \reals^2 \big|
x^2+y^2\le 2\}$. Compute $\displaystyle \int_{\partial D}
x^2dx+y^4dy$. {\small HINT: Use Green's formula.} \end{problem}
\begin{problem}[2 points] Compute $d\omega$ where $\omega=ydx+xdy$.
Let $\displaystyle D= \{(x,y)\in \reals^2 \big| x^2+y^2\le 2\}$.
Compute $\displaystyle \int_{\partial D} ydx+xdy$. \end{problem}
\begin{problem}[2 points] Let $\displaystyle D= \{z \in \comp\big|
|z|\le 1\}$. Compute $\displaystyle \int_{\partial D}
\frac{dz}{4z^2-1}$. {\small HINT: Note that \ \ \
$\displaystyle\frac{1}{4z^2-1}=
\frac{1}{4}\left(\frac{1}{z-\frac{1}{2}}-\frac{1}{z+\frac{1}{2}}\right).
$} \end{problem}
\begin{problem}[3 points] Find the Taylor series around $0$ of
$\sqrt{1-z}$. What is the radius of convergence? \end{problem}
\hspace*{-0.5in} {\bf Problems due October 16:} In problems \ref{aa},
\ref{ab}, \ref{ac} we cover $\pn 1$ with the two usual coordinate
patches, $z\to [z,1]$ and $w\to [1,w]$ with $w=1/z$ on the overlap.
\begin{problem}[2 points]\label{aa} If $p(z)$ is the polynomial
$z^6+z^3+1$ what is the form of the rational `function' it gives rise
to expressed in terms of $w$, i.e., $p(1/w)$? Regarded as a `function'
on $\pn1$ how many zeroes does $p(z)$ have and how many poles does it
have? \end{problem}
\begin{problem}[3 points]\label{ab} If $r(z)$ is the rational
`function' $\displaystyle\frac{z^6+z^3+1}{z^7+8z}$ what is the form of
the rational `function' it gives rise to expressed in terms of $w$,
i.e., $r(1/w)$? Regarded as a `function' on $\pn1$ how many zeroes
does $r(z)$ have and how many poles does it have? \end{problem}
\begin{problem}[3 points]\label{ac} If $p(z)\dr z$ is a one-form with
$p(z)=z^2+1$ what is the form of the rational one-form it gives rise to
expressed in terms of $w$? Regarded as a one-form on $\pn1$ how many
zeroes does $p(z)\dr z$ have and how many poles does it have?
\end{problem}
\begin{problem} [2 points] Given the map $A :\comp\to \comp$ given by
$w= A(z)= z^3$, what is the pullback $A^*dw$? What is the pullback
$A^*(\frac{\dr w}{w})$. \end{problem}
\newpage
\hspace*{-0.5in} {\bf Problems due October 30:}
\begin{problem}[4 points] What is the partial fraction decomposition
over the real numbers $\reals$ of \begin{enumerate} \item
$\displaystyle\frac{x+3}{x^2+1}$? \item
$\displaystyle\frac{x^2+3}{x^2+1}$? \item
$\displaystyle\frac{x^3+x^2+3}{x^2+1}$? \item
$\displaystyle\frac{x^2+3}{(x^2+1)^2}$? \end{enumerate} \end{problem}
\begin{problem}[4 points] What is the partial fraction decomposition over the
complex numbers $\comp$ of
\begin{enumerate}
\item $\displaystyle\frac{x+3}{x^2+1}$?
\item $\displaystyle\frac{x^3+x^2+3}{x^2+1}$?
\item $\displaystyle\frac{x^3+x^2+3}{(x^2+1)^2}$?
\item $\displaystyle\frac{x^2+3}{(x-1)^2(x+3)}$?
\end{enumerate}
\end{problem}
\begin{problem} [3 points] What is the divisor of
\begin{enumerate}
\item $\displaystyle\frac{x+3}{x^2+1}$?
\item $\displaystyle(x-1)^4(x-2)^3(x-7)^8$?
\item $\displaystyle\frac{(x-1)^4(x-7)^4}{(x+1)^4(x+7)^8}$?
\end{enumerate}
\end{problem}
\hspace*{-0.5in} {\bf Problems due November 6:} I strongly recommend you use a
symbolic processing program for these problems. Set up the appropriate matrix
for each problem. Compute the determinant and interpret the answer.
\begin{problem} [3 points] Compute the discriminant of $x^3+ax+b$.
\end{problem}
\begin{problem} [3 points] Compute the resultant of $x^2-ax+b$ and
$x^2-Ax+B$. \end{problem}
\begin{problem} [4 points] Compute the resultant of $x^3-ax+b$ and
$x^2-Ax+B$. \end{problem}
\begin{problem} [2 points] What is the Euler characteristic of a figure
eight? \end{problem}
\hspace*{-0.5in} {\bf Problems due November 13}
The next two problems give a standard example of an integral domain
that is not a UFD.
\begin{problem} [3 points]\label{UFD1} In the domain
$\displaystyle\zed[\sqrt{-3}]$, the norm of an element $\displaystyle
a+b\sqrt{-3}$ is given by $$N(a+b\sqrt{-3})=(a+b\sqrt{-3})\cdot
(a-b\sqrt{-3})=a^2+3b^2.$$ \begin{enumerate} \item Show that
$\displaystyle N((a+b\sqrt{-3})\cdot (c+d\sqrt{-3}))=
N(a+b\sqrt{-3})\cdot N(c+d\sqrt{-3})$. \item Show that
$N(a+b\sqrt{-3})=1$ if and only if $a=\pm 1 $ and $b=0$. \item Show
that $N(a+b\sqrt{-3})=2$ is impossible. \end{enumerate} \end{problem}
\begin{problem} [3 points] In this problem you can use the results of
Problem \ref{UFD1}. \begin{enumerate} \item Show that any element
$x+y\sqrt{-3}\in \zed[\sqrt{-3}]$ with $N(x+y\sqrt{-3})=4$ is
irreducible, i.e., show there are not two elements of $\displaystyle
a+b\sqrt{-3}, c+d\sqrt{-3}\in \zed[\sqrt{-3}]$, which are both not
units with $\displaystyle x+y\sqrt{-3}=(a+b\sqrt{-3})\cdot
(c+d\sqrt{-3})$. {\tt (Hint: If $\displaystyle
x+y\sqrt{-3}=(a+b\sqrt{-3})\cdot (c+d\sqrt{-3})$, then $\displaystyle
4=N(x+y\sqrt{-3})=N(a+b\sqrt{-3})\cdot N(c+d\sqrt{-3})$, and thus
either $N(a+b\sqrt{-3})=1, N(c+d\sqrt{-3})=4$; or $N(a+b\sqrt{-3})=2,
N(c+d\sqrt{-3})=2$; or $N(a+b\sqrt{-3})=4, N(c+d\sqrt{-3})=1$. By the
above either $a+b\sqrt{-3}$ or $ c+d\sqrt{-3}$ is a unit.)} \item Show
that $4=2\cdot 2=(1+\sqrt{-3})(1-\sqrt{-3})$. Why does this show that
$4$ can not be uniquely factored, and in particular that
$\displaystyle\zed[\sqrt{-3}]$ is not a UFD. \end{enumerate}
\end{problem}
I strongly recommend you use a symbolic processing program for the
next problem. Set up the appropriate matrix and compute its
determinant.
\begin{problem} [4 points]Using the resultant, eliminate $z$ between
the two cubics $x^3+y^3+z^3=0$ and $x^2y+xyz+yz^2+z^3=0$. If you
could factor resulting polynomial into factors that are linear in
$x$ and $y$, how could you use this to solve find the nine common
zeroes of the above two cubics? \end{problem}
\newpage
\section{First Test, October 2, 1996}\setcounter{problem}{0}
\subsection*{Problems}In the following you must show your work. All $\pn 1$'s and $\pn 2$'s are over the
complex numbers unless explicitly stated otherwise.
\medskip
\begin{problem}[12 points total]We have the map $\phi:\comp^2\to \pn
2$ given by $\phi(x,y)= [x,y,1]$.
\begin{enumerate}
\item Which points of $\pn 2$ are not in the image under $\phi$ of
$\comp^2$?
\item Is the map $\phi$ one-to-one on $\comp^2$?\
\end{enumerate}
\end{problem}
\medskip
\begin{problem}[8 points]What homogeneous polynomial $p(x,y,z)$ has the following partial derivatives:
$$
\pard{p}{x}=4x^3+3zx^2,\ \ \
\pard{p}{y}=4y^3,\ \ \
\pard{p}{z}=x^3.$$
\end{problem}
\medskip
\begin{problem}[9 points] Let $f(x,y)=x^3+y^2+x=0$ define a curve in
$\comp^2$. Considering $\comp^2$ as a subset of $\pn 2$ by the map
$(x,y)\to [x,y,1]$, what is the homogeneous polynomial of the same
degree as $f$ which gives the equation $f$ on $\comp^2$?
\end{problem}
\medskip
\begin{problem}[10 points total] We have the curve $C$ defined by
$x^3+y^2z=0$.
\begin{enumerate}
\item What are the singular points of $C$?
\item How many
distinct singular points are there for this equation?
\end{enumerate}
\end{problem}
\medskip
\begin{problem}[12 points total] State B\'ezout's theorem. Is there
any conflict with the fact that the curves in the {\bf real} projective
plane $\pn 2_\reals$ defined by $x^2+y^2-z^2=0$ and $x^2+y^2-4z^2=0$
have no points in common? \end{problem}
\medskip
\begin{problem}[15 points total] Let $C$ be the curve in $\pn 2$ defined by $p(x,y,z)=yz^3-x^4+2x^2z^2-z^4=0$.
\begin{enumerate}
\item What is the homogeneous equation of the tangent line $L$ of the
curve $C$ in $\pn 2$ at $[1,0,1]$? (SOME HELP: $\displaystyle
\pard{p(1,0,1)}{x}=0, \pard{p(1,0,1)}{y}=1, \pard{p(1,0,1)}{z}=0$.)
\item What points does this line $L$ meet $C$ in? What are their
multiplicities?
\item Why is or is not this count of the number of points of
intersection of $L$ and $C$ compatible with B\'ezout's theorem?
\end{enumerate}
\end{problem}
\medskip
\begin{problem}[12 points]Find the automorphisms
$\displaystyle\phi(z)=\frac{az+b}{cz+d}$ of $\pn 1$ that take $\infty$
to $\infty$. Which of these also take $0$ to $0$? \end{problem}
\medskip
\begin{problem}[10 points total]What is the power series expansion of
$\displaystyle\frac{1}{z}$ around $z=1$? What is the radius of
convergence of this power series? \end{problem}
\medskip
\begin{problem}[12 points total]Compute the following line integrals
over the unit circle, $|z|=1$. (A convenient parameterization of the
unit circle is given by $z(t)=e^{it}$ for $0\le t\le 2\pi$.)
\begin{enumerate}
\item $\displaystyle \int_{|z|=1} z{\rm d} z$.
\item $\displaystyle \int_{|z|=1} \oline z{\rm d} z$.
%\item $\displaystyle \int_{|z|=1} \frac{(1+z^2){\rm d} z}{z-\frac{1}{2}}$.
\end{enumerate}
\end{problem}
\newpage
\section{Second Test, November 15, 1996}\setcounter{problem}{0}
\begin{problem}[6 points] If $r(z)$ is the rational `function'
$\displaystyle\frac{z^3}{z^2+8}$ what is the form of the rational
`function' it gives rise to expressed in terms of $w$, i.e., $r(1/w)$?
Regarded as a `function' on $\pn1$ how many zeroes does $r(z)$ have and
how many poles does it have? \end{problem}
\medskip
\begin{problem}[6 points]\label{sixteen} If $p(z)\dr z$ is a one-form
with $p(z)=z^4$ what is the form of the rational one-form it gives rise
to expressed in terms of $w$? Regarded as a one-form on $\pn1$ how
many zeroes does $p(z)\dr z$ have and how many poles does it have?
\end{problem}
\medskip
\begin{problem}[12 points] What is the partial fraction decomposition
over the real numbers $\reals$ of
\begin{enumerate}
\item $\displaystyle\frac{5x+3}{(x^2+5)^2}$?
\item $\displaystyle\frac{5x+3}{x^2+1}$?
\item $\displaystyle\frac{x^3+x+3}{x^2+1}$?
\end{enumerate}
\end{problem}
\medskip
\begin{problem}[16 points] What is the partial fraction decomposition
over the complex numbers $\comp$ of
\begin{enumerate}
\item $\displaystyle\frac{x^2}{x-1}$?
\item $\displaystyle\frac{x}{x^2+1}$?
\item $\displaystyle\frac{x^3-x+3}{x^2-1}$?
\item $\displaystyle\frac{x+1}{(x-1)^2}$?
\end{enumerate}
\end{problem}
\medskip
\begin{problem} [12 points] What is the divisor on $\pn 1$ of
\begin{enumerate}
\item $\displaystyle\frac{x}{x^2-1}$?
\item $\displaystyle(x-2)^3(x-4)^2(x-7)$?
\item $\displaystyle\frac{(x-1)^4(x-3)^4}{(x+2)^2(x+9)^4}$?
\end{enumerate}
\end{problem}
\medskip
\begin{problem}[10 points]You are given two real polynomials $p(x) =
x^2-1$ and $q(x)=(x-1)^3$.
\begin{enumerate}
\item Find the greatest common divisor of $p(x)$ and $q(x)$.
\item Find the resultant of these two polynomials---an explicit numerical
answer and a justification for it are required.
\end{enumerate}
\end{problem}
\medskip
\begin{problem} [6 points] Set up the matrix (i.e., you do not have to
compute the determinant) to compute the discriminant of
$x^3+2x^2+3x+4$. \end{problem}
\medskip
\begin{problem} [6 points] Compute the resultant of $x^2+1$ and
$x^2+2$. (For this problem you must compute a determinant. You must
show your work, i.e., no credit will be given for simply writing down
the determinant). \end{problem}
\medskip
\begin{problem} [6 points] You have a polynomial $p(x,y)\in
\comp[x,y]$. You know that there is no point $(x,y)\in \comp^2$
with $p(x,y) = 0$. Moreover you know $p(0,0)=1$. Write down all such
$p(x,y)$ explicitly. Justify your answer. Show the importance in the
above of considering all complex solutions by giving an example of a
nontrivial polynomial that has no zeroes on $\reals^2$. \end{problem}
\medskip
\begin{problem} [10 points]The resultant needed to eliminate $y$
between the $1+x^3+y^2+y^3=0$ and $x^2+xy+y^2+y^3=0$ is
$p(x)=1+x-3\,x^{2}+2\,x^{3}+5\,x^{4}-5\,x^5+x^{6}+4\,x^{7}-3\,x^{8}+x^{9}$.
Assume that you know how to find the complex solutions to any one
variable polynomial with complex coefficients, e.g., $p(x)=0$. Explain
(with a justification) how you might find the
common zeroes in $\comp^2$ of $1+x^3+y^2+y^3=0$ and
$x^2+xy+y^2+y^3=0$. (Do not worry about multiplicities of solutions.)
\end{problem}
\medskip
\begin{problem} [10 points] What is the Euler characteristic of the
Nephroid of Freeth? \end{problem}
\begin{center}
\tt A picture of the nephroid of freeth was included here
% \includegraphics[scale=0.6]{nephroid.eps}
\end{center}
\end{document}