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\centerline{MATH 392 -- Seminar in Computational Commutative Algebra}
\centerline{Information for Midterm Exam}
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{\it General Information}
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Still to be determined is whether the exam will be given in class
on Friday, March 22 or the previous Thursday evening.  Either 
option is fine with me; we will discuss this in class on 
Wednesday, March 13.  
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\item{$\bullet$} If we do the exam Friday, you will have the full period. 
If we do it Thursday evening, you will have two hours.  I reserve the right
to make the exam slightly longer (nowhere near twice as long, though) 
if we decide to do it in the evening.
\item{$\bullet$}  This is an individual exam.  No 
sharing of information with others in any form will be permitted while
you are taking it.
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\item{$\bullet$} You may use a calculator during the exam,
but {\it no other electronic devices}.
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\item{$\bullet$} There will be four or five
problems (each possibly with a few separate parts).  These
questions will draw on {\it the material we have studied}
from Chapters 1, 2, and 3 of our text.  Some 
sample exam questions are given later in this document. 
\item{$\bullet$} I won't ask you to do lots of computations by hand because we have done
almost all of that heavy lifting using Maple, and you won't
have that available during the exam.  {\it One exception to that
general rule is that you should be prepared to do ``reasonable'' polynomial divisions
in one or several variables, and/or carry out the Euclidean Algorithm to 
find the gcd of two polynomials in one variable.}  
\item{$\bullet$} The exam will focus
more on the conceptual background, definitions, key results we
have studied, and perhaps understanding what output from Maple for
some of the computations we have done means.  I may ask you for statements and/or
proofs of main results.  If a question says ``state Theorem X'' then you need only give
the statement of the Theorem; there is no need to give the proof.
Any such question will be the same as one of the parts of the practice problems below.
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{\it Some sample practice/review problems}
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I.  Draw or describe in words each of the following varieties.
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\item{A)}  ${\bf V}(x^2 + y^2 - 4, x - y + 1)$ in ${\bf R}^2$
\item{B)}  ${\bf V}(z - xy)$ in ${\bf R}^3$
\item{C)}  ${\bf V}(x^2 + y^2 - 9, z - x^2)$ in ${\bf R}^3$.
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II.  All parts here refer to the polynomial ring $k[x_1,\ldots,x_n]$.  
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\item{A)}  Define:  $I$ is an ideal.
\item{B)}  Define:  The radical $\sqrt{I}$ of an ideal $I$.
\item{C)}  Show that if $I$ is an ideal, then $\sqrt{I}$ is also an ideal.   
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III.  
\item{A)}  Prove that every ideal $I$ in the polynomial ring $k[x]$ (one variable)
is principal (that is, $I = \langle g(x)\rangle$ for some single polynomial).
\item{B)}  Find $g(x)$ as in part A for the ideal $I = \langle x^3 + 4x - 16, x^2 - 5x + 6\rangle$
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IV.  
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\item{A)}  Define:  $>$ is a monomial order on $k[x_1,\ldots,x_n]$.
\item{B)}  What property of a monomial order implies that the division algorithm 
in $k[x_1,\ldots,x_n]$ will always terminate in a finite number of steps?  (That is, the
algorithm terminates no matter what the divisors $f_1,\ldots,f_s$ and the dividend $f$ 
are.)
\item{C)}  What are the quotients and the remainder on division of 
$f = x^3 y - 4y$ by $f_1 = x^2 + x y$ and $f_2 = y - 1$ using the lex order with $x > y$?
Using the lex order with $y > x$?
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V.  
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\item{A)}  State Dickson's Lemma.
\item{B)}  Define:  $G$ is a Gr\"obner basis for an ideal $I$ with respect to a monomial
order $>$.
\item{C)}  Prove that Gr\"obner bases exist for every ideal $I$
in $k[x_1,\ldots,x_n]$ and with respect to every monomial order.  (Hint: Use Dickson's Lemma; 
state the definition of any auxiliary ideals you need in the proof.)
\item{D)}  Deduce from C that every ideal in $k[x_1,\ldots,x_n]$ has a finite
basis (the Hilbert Basis Theorem).  
\item{E)}  Is $\{x^2 y - xy + 1, xy^2 + 1\}$ a Gr\"obner basis for the ideal generated
by these two polynomials?  Why or why not?  State any key results you are using to 
derive your conclusion.
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VI.  
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\item{A)}  Show that if $G$ is a Gr\"obner basis for an ideal $I$ with respect 
to the monomial order $>$ and $f \in I$, then 
the remainder on division of $f$ by $G$ using $>$ is {\it zero}.  Do your proof
as a formal proof by induction on an integer $m$ counting the steps in the division.
\item{B)}  Define:  The $\ell$th elimination ideal $I_\ell$ of an ideal $I \subseteq k[x_1,\ldots,x_n]$.  
\item{C)}  State and prove the {\it Elimination Theorem}.
\item{D)}  State the {\it Extension Theorem}.
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VII.  For all parts of this question, refer to the Maple output from the course homepage.
\item{A)}  From the computation for this part, what can you conclude about $f = x^3+y^3+z^3- 2z^2 + 2y - 33z$?
\item{B)}  From the computation for this part, determine Gr\"obner bases for the elimination
ideals $I_1$ and $I_2$ for $I = \langle y^3+z^3+x^2-1, -y^2+x+z+1, x^2-y^2-z+1\rangle$.
\item{C)}  Continuing from part B, what does the Extension Theorem tell you about
points in the variety ${\bf V}(I)$ in ${\bf C}^3$?  Explain step by step and say how many
points there are on the variety for each $z$ solving the equation $BB[1] = 0$.  Note:  A consequence
of the Fundamental Theorem of Algebra is that there are 10 such values of $z$ -- two real ones and
four complex conjugate pairs.
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