Homework for Math 676: Computational Algebraic Geometry
Spring 2009 (Bates)
I would like everybody to turn in 20 points worth of homework by the end
of the semester. Problems will range from very easy (1-2 points) to very
difficult (9-10 points). There will be plenty of homework assigned (both
easy and hard), so don’t worry too much if you have trouble with some
problems.
1. [2 points] We had quite a discussion about the definition of a primary
ideal. The definition I gave in class is fairly standard:
An ideal I ⊂ F[x1 , . . . , xn ] is primary if f g ∈ I =⇒ f ∈ I
or gm ∈ I for some positive integer m.
On the basis that we are working over commutative rings, it was asked
whether this is equivalent to
An ideal I ⊂ F[x1 , . . . , xn ] is primary if f g ∈ I implies that
either f k ∈ I for some positive integer k or gm ∈ I for some
positive integer m.
Your job is to either prove that these two characterizations are equivalent or produce an example that shows that they are not.
2. [3 points] Although we will not use the dictionary between algebra
and geometry too deeply or too often (at least early in the course),
there are some very nice connections between the two areas. If you
haven’t seen this material before, this would be a good problem for you.
Let I = (f1 , . . . , fk ) and J = (g1 , . . . , gm ) be ideals in a polynomial
ring F[x1 , . . . , xn ]. (Each part is worth 1 point.)
• Show that V (I + J) = V (I) ∩ V (J), where I + J = {f + g : f ∈
I and g ∈ J}.
• Show that V (I·J) = V (I)∪V (J), where I·J is the ideal generated
by all products f · g with f ∈ I and g ∈ J.
• Show that V (I ∩ J) = V (I) ∪ V (J). You may assume that I · J ⊂
I ∩ J.
(Hint: §4.3 of the [CLO1] should be handy, but please write up your
solutions in your own words....)
3. Write a short procedure to do Sturm sequences and bisection on a
univariate polynomial p in a given interval [a,b]. (Notice that there
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are Maple commands ”sturm” and ”sturmseq.”)
(a) [2 points]
√ Make
√ the procedure find one root of f (x) = (x −
π)(x − 2)(x − 3) with 10 digits of accuracy.
(b) [1 point] Make it find all roots of f (x) with 10 digits of accuracy.
4. (a) [2 points] Have Maple generate 10 polynomials with rational
coefficients which do not have GCD 1. Form a random element
f of the ideal that they generate (by taking a combination of
the original 10 polynomials with random polynomial coefficients).
Use Maple’s GCD routine to find a generator for your ideal and
test whether f is in your ideal.
(b) [1 point]Now try larger examples (larger coefficients, higher degree, more polynomials, etc.) and try to get Maple to take more
than one minute to solve such a problem.
5. [2 points] Cook up your own weighted (matrix) term order. Find a
single polynomial that has a differnet leading term depending on your
choice of lex, grlex, grevlex, or your weighted term order.
6. [4 points] Write a script to test the likelihood that a randomly chosen set of generators is a Gröbner basis for the ideal they generate. In
particular, generate 1000 ideals, each consisting of 3 randomly chosen
polynomials in 3 variables (small integer coefficients and degree no
more than 5). For each ideal, check whether or not the chosen generators form a Gröbner basis for the ideal (using the criteria discussed in
class). Keep track of how many of each type you see, and, in the end,
print out the number of them that were indeed Gröbner bases and the
number that were not. Please respond briefly (i.e., is this surprising
or can you reason it out?).
7. [8 points] For 5 points, implement Buchberger’s original algorithm.
In particular, implement the first algorithm that we discussed, without
any of the bells and whistles that can be added on. Your implementation can be Maple (or otherwise, as you like) and may use both the
Groebner package only for extracting leading terms and such. I’ll give
an extra point if you incorporate Lemma 3 on page 89 of [CLO1] (to
shrink the basis a bit). For three extra points, call your implementation from a loop that forms all elimination ideals of a given ideal and
returns at least one point on the variety of the original ideal.
8. [3 points] For 2 points, find all solutions of the ideal < x2 + y 2 + z 2 −
4, x2 + 2y 2 − 5, xz − 1 > using elimination/extension with lex(x,y,z)
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(using Maple, if you like). Now reverse the variable order and do it
again for the other point.
9. [2 points] The twisted cubic curve is a smooth curve in C3 and is
one of the canonical examples in algebraic geometry. Its points are
given by the parameterization (t, t2 , t3 ), and one set of generators is
< y − x2 , z − x3 , z 2 − y 3 >. Use elimination theory to recover the
parameterization.
10. [5 points] The ideal quotient of two ideals is a handy operation that
we have not yet discussed in class. Here’s a definition: For ideals
I, J ⊂ F[x1 , . . . , xn ], the ideal quotient of I with J (a.k.a., I “colon”
J) is the set I : J = {f ∈ F[x1 , . . . , xn ]|f · J ⊂ I}, i.e., the set of
polynomials that push all of J into I. (NOTE: The obvious analogous
definition works for general commutative rings).
(a) For 2 points, prove that I : J is an ideal of F[x1 , . . . , xn ].
(b) For another point, what is I : J if J ⊂ I?
(c) The ideal quotient has geometric value – it allows us to do some
sort of subtraction on varieties. In particular, V (I : J) = V (I) − V (J),
where Z is the Zariski closure of the set Z. Proving this will get
you two more points. (Recall that the Zariski topology has as
closed sets all algebraic sets.)
11. [6 points] The family of cyclic n-roots problems is a popular benchmark in certain circles. A fun fact is that the associated variety is
zero-dimensional for prime n. For composite n, the variety is equidimensional and the number of irreducible components is known and
interesting (though the degrees of the components are trickier). Jan
Verschelde has a great catalog of polynomial systems on his website
(http://www.math.uic.edu/ jan/) – click on Demos(tree), cyclic nroots, and original to see the systems for n = 5..11. For one point, get
Maple, Singular, or Macaulay 2 to confirm that the case n = 5 is indeed zero-dimensional. For another point, display all 70 solutions. For
two other points, do the primary decomposition in the case n = 6, i.e.,
give me generators and the dimension for each primary component.
For two more, do the same for n = 8 (WARNING: n = 8 could take
some time). The varieties of these ideals would make for an interesting
talk – check out the references on Verschelde’s website, particularly for
n = 7, 8, 9, 10.
12. [3 points] In lab 5, #2 under “known positive-dimensional” has a
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nice little (positive-dimensional) solution set (variety). Get Maple,
Macaulay 2, and Singular to list the components (generators for the
corresponding ideals) – one point each.
13. [1 point] Show that the equations defining the twisted cubic curve do
indeed yield a unique (irreducible) curve of degree 3.
14. [5 points] Use Bertini or PHC to compute the numerical irreducible
decompisition of any of the polynomial systems from the previous three
problems – one point per problem.
15. [points?] Suggest your own homework problem! If you been tinkering
with some idea/problem/software package and discovered something
nontrivial, it may count for homework credit. Let me know, and I’ll
try to assign some nonnegative (you hope!) number of points for the
problem. The point of the homework is to force you to get your hands
dirty...I don’t particularly care which kind of dirt it is.
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