Lab 5

Exercise 1: Solving Polynomial and Nonlinear Equations

  1. Solve the following nonlinear system of equations:
    [ x^2 + y^2 - z = 0
    xy - z^2 = 1
    z^3 - x + y = 2 ]
  2. Analyze whether all solutions satisfy (x, y, z \in \mathbb{R}).

Exercise 2: Gröbner Bases

  1. Define the ideal (I = \langle x^2 + y^2 - 1 = 0, \quad x^2 - z = 0, \quad y^2 - z^2 = 0 \rangle) in (\mathbb{Q}[x, y, z]).
  2. Compute a Gröbner basis for (I) using:
    • Lexicographic order: lex
    • Graded lexicographic order: grlex
    • Graded reverse lexicographic order: grevlex
  3. Compare the results:
    • List the elements of the basis for each order.
    • Analyze the differences.
  4. Determine whether the ideal (I) is zero-dimensional and explain what this implies about the solutions of the system.

Exercise 3: Testing Membership in Ideals

  1. Given the ideal (I = \langle x^3 - y^2, x^2y - z^3 \rangle), check whether the polynomial (f = x^3y^2 - z^5) belongs to (I).
  2. Justify your result using the properties of Gröbner bases.

Exercise 4: Exploring quotient rings

  1. Define symbols s, c and then define a quotient ring using
    K = QQ.old_poly_ring(s, c) / [s**2 + c**2 - 1] Then enter
p = K.to_sympy
c = K.convert(c)
s = K.convert(s)
  1. Explain with your own words the results you get for the followings:
p(c**2)
p(s**2)
p(s**2 + c**2)
  1. Experiment with converting elements between various rings.

Exercise 5: Operations on Ideals

  1. Warm-up task: what is the intersection of the principal ideals (6) and (8) in ZZ?
  2. Construct the ideals (I_1 = \langle x^2 - y^2 \rangle) and (I_2 = \langle x^3 - y^3 \rangle) in (\mathbb{Q}[x, y]).
  3. Compute the following:
    • (I_1 + I_2) (sum of ideals).
    • (I_1 \cdot I_2) (product of ideals).
    • (I_1 \cap I_2) (intersection of ideals).
  4. Verify the containment relationships:
    • Is (I_1 \subseteq I_2)?
    • Is (I_2 \subseteq I_1)?