Math 4800/6080. Week Four Starter Problem
1. Find examples of quadratic polynomials:
p(x, y) = ax2 + bay + cy 2 + dx + ey + f with at least one of a, b, c 6= 0
whose level sets (p(x, y) = 0) ⊂ R2 are each of the following:
(a) A hyperbola
(b) A parabola
(c) An ellipse
(d) A pair of intersecting lines
(e) A pair of parallel lines
(f) A single line
(g) A single point
(h) The empty set
2. What is the fundamental “topological” difference between:
(a) The circle in RP2 that is the completion of a line, and
(b) The circle in RP2 that is the completion of an ellipse/parabola/hyperbola?
3. Classify all the topological types of completed level sets:
(P (x : y : z) = 0) ⊂ RP2
where P (x : y : z) = ax2 + bxy + cy 2 + dxz + eyz + f z 2 and at least
one of the coefficients is not zero.
4. Find as many different topological types as you can for the level sets
of cubic polynomials:
p(x, y) = ax3 + bx2 y + cxy 2 + dy 3 + ex2 + f xy + gy 2 + hx + iy + j
where at least one of a, b, c, d is non-zero. Then do the same for the
completions (actually, you might do the completed level sets first).
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