Spring 2019
Math 648: Computational Algebraic Geometry -- Winter 2019

Homework

1. Use resultants to compute the implicit equation of the parametrized curve, x = t³+3t²-2 y = t³-2t
2. Problem 4, Section 2.1. Prove the equality of the two formulas for the Discriminant in Example 2.1.5. Warning: the LHS should be divided by f₀, and not multiplied by it, as it is in the current version on line.
3. Let f = 5x²y + 7xy² + 2     and g = 13x⁴ + 17x²y² + 11x   be polynomials in Q[x,y] with the degree reverse lexicographic term order. In one of the computer algebra systems, compute the Gröbner basis of the ideal they generate, implementing the steps in the Buchberger algorithm. This is started in either of 7Feb.m2 or 7Feb.sing. It is your choice which one to use.
4. Problem 3, Section 2.3. This gives an elementary iprovement to Buchberger's Algorithm.
5. From Section 2.4, do problems 2, 5, and 8. The Singular script sparse.sing (that I ran in class) may help, as may sparse_3.sing.
   Here are Macaulay2 versions, due to Thomas Yahl: sparse.m2 and sparse_3.m2.

1. Section 1.2, Numbers 3 and 10.
2. Section 1.3, Number 5.
3. Section 2.2, Numbers 2 and 6.