Spring 2019
Math 685: Computational Algebraic Geometry -- Winter 2019 (Section 700)

Homework

1. Warm up. Try to do the experiment in Exercise 5 of Section 2.4 of my notes. This is to see the conclusion of Bézout's Theorem in practice. For this n should be 1,2,3, or 4, and the degree should be not so large. You will see that when the expected number of solutions is largish, the computation is very slow. We may need to discuss/find a way to easily generate random polynomials in n variables of a given degree.
2. One experiment. Try now Exercise 6 for n= 1, 2, 3, and 4. You might set up n=5, or n=6, but it may not run unless you work over a finite field. We may need to discuss what happens in positive characteristic.
3. Sparse polynomials. Exercise 7 is a wide open exploration. Using the scripts that I provided (or your own scripts), do many computations with different exppnent matrices A, in the case of two variables. Here is some graph paper 1/2 inch and 1/4 inch. I suggest that you plot the exponent vectors (columns of A and compare those data to the number of solutions that you find.
   This exploration may continue into higher dimensions n=3, or where the two polynomials in the system have different monomials.

1. Exercises 1, 2 (the Whitney umbrella), 3, and 4. (For all of these, give code. We may need to discuss how to determine the number of real solutions in Problem 4).

1. Exercises 1,2,3,5,7. (1 & 2 are similar and you can reuse code.)

1. Please do exercise 1, checking that the remaining S-polynomials reduce to zero in Example 1. Note that most of them were checked in earlier steps, so you do not need to do 10 reductions.
2. 2 (b), and then do 3 on this one (four Gröbner bases). That is, first compute Groebner bases for the ideal generated by x²+y ,  x⁴ + 2x²y + y² + 3 ,  in lex, grlex, and then give the reduced Gröbner bases in these two orders. I think that you can take x>y.
3. 8. For this, describe an algorithm whose input is a Gröbner basis for an ideal and output is the reduced Gröbner basis. Include a proof of correctness.
4. Here is an interesting exercise. Compute both a grlex and a lex Gröbner basis for the ideal with the three generators
   xy²+xz² + x-1  ,  yz²+yx² + y-1  ,  zx²+zy² + z-1  .
   Repeat this with the exponents 2 changed to 3. What are the dimensions and degrees of these ideals? In Singular, to make sure that you get a reduced Gröbner basis, include the line near the top of your file option(redSB);. What do you notice about this? You will want to submit the output files.

1. Please do exercises 2, 3, 5, and 10. Only 3 is a proof, the others are computations.

1. Section 2.5, Exercises 1 and 2. These explore when you do not have a Gröbner basis for an ideal.
2. Section 2.5, Exercise 5. This is to think more about the definition of a Gröbner basis
3. Section 2.5, Exercise 9. This relates Gröbner bases to linear algebra concepts.
4. Section 2.5, Exercise 13.
5. If you want to redo a problem from an earlier problem set for feedback from Frank, please send it to him (his problem with attachments seems to have abated with a software update). Also, please entitle the .pdf with your name, and make sure your name is on what you submit.)

1. Exercises 3 and 4, Section 2.4 in CLO. These are to help you understand monomial ideals and Disckson's Lemma. You may find graph paper helpful.
2. This exercise asks you to do a computational experiment.
   In either Singular or Macaulay2, create the monomial ideal generated by x²y³ and compute its dimension and multiplicity/degree. (These are commands in the language. In Singular, if I is your ideal then I = std(I); dim(I); and mult(I); will do this.)
   In Macaulay2, you can get started with R = QQ[x,y], I=ideal{x^2*y^3}, dim I, and then degree I (probably on separate lines).
   Add the monomials x⁶, xy⁷, y⁸, x³y, and y⁴, one at a time, computing the dimension and multiplicity/degree each time. Compare this to a two-dimensional plot of the monomials in the ideal. Can you make any conclusions from this experiment? If needed, add a few more monomials to the ideal to begin to see what is happening.
   Here is are links to graph paper 1/2 inch and 1/4 inch, if you need them for sketching the ideals.
   The dimension of an ideal is the dimension of its variety, to which Cox, Little, and O'Shea devote a later chapter. Please take this as a black box. It is however easy to define zero-dimensional. An ideal I is zero-dimensional if and only if V(I) is a finite set if and only if k[x₁,...,x_n]/I is a finite-dimensional k-vector space. In that case, the multiplicity of I is its degree (there is no ambiguity in these terms for zero-dimensional ideals), and this is simply the dimension of the ring k[x₁,...,x_n]/I as a k-vector space.
3. Exercises 8 and 9, Section 2.4 in CLO. These should not be so involved, but they show how monomial ideals are much better than ordinary ideals. Here is a statement of them:
   8) A basis for a monomial ideal is minimal if no element in it divides any other (for then the second could be removed without changing the ideal).
           (a) Prove that every monomial ideal has a minimal basis
           (b) Prove that every monomial ideal has a unique minimal basis
   9) If I is a monomial ideal with generators x^α(1),...,x^α(m), prove that a polynomial f lies in I if and only if the remainder of f upon division by x^α(1),...,x^α(m) is zero. There is a hint to use Lemmas 2 and 3.
4. If you want to redo a problem from an earlier problem set for feedback from Frank, please send it to him (his problem with attachments seems to have abated with a software update). Also, please entitle the .pdf with your name, and make sure your name is on what you submit.)

1. Run one of Week6.sing or Week6.m2 and examine the output.
2. Please do Exercise 2.
3. To help understand the division algorithm, do Exercise 4.
4. Do Exercise 5
5. Do Exercise 6
6. If you want to redo a problem from an earlier problem set for feedback from Frank, please send it to him, again at sottile@math.tamu.edu. (There is a problem with attachments on that gmail account. Also, please entitle the .pdf with your name, and make sure your name is on what you submit.)

1. This exercise is to get you running either Macaulay2 or Singular. Taylor and I made two scripts Feb_11.m2 and Feb_11.sing, which are Macaulay2 and Singular (respectively; note the extension) scripts. Please run one of them, and read it. They both do some of the exercises in Chapter I and the beginning of Chapter II. You may not understand what all the commands are, or are doing.
       The last computation in each computes the equation for some crazy plane curve that is given by a parametrization x=f(t) and y=g(t). Copy this part of the code and change the copy, altering the coefficients and degrees (in t) of f and g. What does this do to the resulting polynomial F(x,y) that is computed? Experiment, and be glad that the computer is doing all the work for you.
2. To study monomial orderings, consider monomials in x, y, and z with x > y > z. Write the six monomials of degree two in x, y, z in both graded lexicographic order and in graded reverse lexicographic order. Do the same for the 10 monomials of degree 3 in these variables.
3. For more, do exercises 1 and 2 from Section II.2.
4. Please work to give a proof of Lemma 8, part (a). This will be much more difficult, but it is very important, as it underlies many arguments and algorithms. It is essentially using transitivity of the order and an argument that there is no cancellation at the top. You may find thinking about Exercise 11 is helpful. I would also do an example or two by hand, and maybe even experiment with some Singular or Macaulay2 code (ask for help with that on Piazza). I expect this will be one where we have a lot of back and forth in Piazza, and likely you will hand in a revised version to me in the future. If you can get this, that is great, but I expect to work with some of you on this one.
5. Speaking of revisions, please hand in to Frank (in .pdf, and at sottile@math.tamu.edu) a revised version of one of the problems from Week 3.

1. Problem 1 d. Determine whether or not x³-1 lies in the ideal generated by the following 2 polynomials: x⁹-1 and x⁵+x³-x²-1.
2. Problem 2c. Find a parametrization of the affine variety in either R³ or C³ defined by the equations, y - x³ = 0 and z - x⁵ = 0.
3. Problem 3b. Find implicit equations for the affine variety in either R⁴ or C⁴ parametrized by x₁ = 2t - 5u,   x₂ = t + 2u,   x₃ = -t + u,   x₄ = t+ 3u.
   (This is a topic from linear algebra.)
4. Problem 5. This is somewhat theoretical. Let me give you a hint. The ring k[x,y] of polynomials in two variables is also a vector space over k. A basis for this is given by monomials, x^a y^b, for a,b nonnegative integers. Part a. asks for you to determine the dimension of the space of bivariate polynomials of degree at most m.
   Similarly, the polynomial ring k[t] has a basis of monomials t^a for a a nonnegative integer. When f(t) is a polynomial of degree at most n, (f(t))^a is a polynomial in t of degree at most an, and thus is an element in the vector space of polynomials in k[t] of degree at most an. You should verify/understand that this vector space has dimension an + 1.
   The point of this exercise is that the 'monomials' (f(t))^a(g(t))^b, which are polynomials in t, will necessarily have a linear dependency (when there are more monomials than the dimension of the ambient vector space in k[t]) and that this linear dependency gives a polynomial F(x,y) that vanishes on the parametrized curve.
5. This last problem is special. For each of you, choose a problem from the first homework that you did not get completely right. Taking advantage of Taylor's comments, or anything else, try to revise your answer (or completely redo it) and hand that into me (Frank). Either email it to me at fjsteachmath@gmail.com or type it into Piazza and send it to me as a private note. The purpose of this is for me to work with you to improve your answer via a one-on-one discussion. Both Taylor and I have had similar instruction earlier in our careers, and it was very helpful. I hope this will also work for each of you.

1. Problem 2, involving the Vandermonde determinant, is a very nice application of of Corollary 3.
2. Problem 4, While this can be done directly, it is much easier to use Proposition 6 (ii).
3. Problem 9. It was not hard (but only slightly tedious) to compute the gcd of the three polynomials by hand. Later, we will use software for such computations.
4. Problem 11, which solves the consistency problem for univariate polynomials.

1. Prove Proposition 4, that if f₁, f₂, ..., f_s and g₁, g₂, ..., g_t both generate the same ideal J, then they define the same variety, V(f₁, f₂, ..., f_s) = V(g₁, g₂, ..., g_t) .
2. Do Problem 8 on radical ideals. These will be important near the end of the course when we prove basic theorems in algebraic geometry.
3. Please try problem 12, it might be a bit challenging. Its main part will be to determine the ideal of the space curve parametrized by the monomials (t²,t³,t⁴). So that we can discuss this, use the coordinates (x,y,z).
4. Also do 15 (a), showing that for any subset S of kⁿ, the set of polynomials vanishing identically on S forms an ideal.

1. Prove that the set Z of integers is not a subvariety of the set C of complex numbers.
2. Problem 6 in the Exercises for Section I.2: Show that every finite subset of kⁿ is an affine variety.
3. Problem 8 in the Exercises for Section I.3: This walks you through the parametrization of a nodal cubic, y²=cx²-x³ by nonvertical lines through the origin. It is similar to a problem I assign in Math 629.