Examples and Exercises

Question 1:

1. Solve the following system using the black-box solver.
   

   $$x^2+4y^2-4 =$$ 0

   $$2y^2-x =$$ 0

   
   
2. Draw both curves with a software of your choice (e.g. Maple) and check if the real solutions you found with phc make sense.
3. Can you change the constants of the parabola such there are 4 real intersection points? Verify this with phc.

Question 2: Consider the following polynomial.

$$x\: \prod_{j=1}^{m-1} (x^2-j)$$

1. How many roots do you expect? Why is this an extremal example regarding Descartes' Rule of Signs?
2. Compute the roots for $m=3$ using phc -b. (The output will be slightly different from what we described above, but since it's short you will find the information you need.)

Question 3:

1. Solve the system
   

   $$x^{108} + 1.1 y^{54} - 1.1 y =$$ 0

   $$y^{108} + 1.1 x^{54} - 1.1 x =$$ 0

   
   using the black-box solver. (This may take a few minutes.)
2. While phc is working for you, look up Kouchnirenko's Conjecture and apply it to the above system.
3. Compare the prediction of the Conjecture and the number of solutions that phc found. See [1] fro details.

Question 4:

1. Solve the system
   

   $$x_1^2+x_2-3 =$$ 0

   $$x_1+0.125 x_2^2-1.5 = 0 .$$

   
   
2. Give the multiplicity of the roots and say if they are regular or singular. (You will have to look at the long output file to see that.)
3. Compute the Jacobian of the system and verify the singular solutions.

Question 5: Consider the following graph with prescribed edge lengths

We fix the first and second vertex as $(x1,y1)=(1,1)$ and $(x2,y2)=(3,1)$ . Furthermore we give the following values for the edge lengths.

$$L_1=2,\quad L_2=0.9 ,\quad L_3=1.3 ,\quad L_4=1.5 ,\quad L_5=0.7 ,\quad L_6=2.7 ,\quad L_7=1.9$$

1. Model this situation as a system of polynomial equations.
2. Use phc -m to compute the mixed volume of this system.
3. Use phc to compute the solutions to this system. How many do you get? Does this agree with the mixed volume?
4. Look up Bernshtein's second theorem (see the courses homepage if you can't find it anywhere else).
5. According to the theorem there has to be a direction $w$ such that $\partial_w f=0$ has a solution in $({\mathbb{C}}^*)^6$ . Find such a $w$ .

Question 6: Cut the unit sphere in ${\mathbb{R}}^3$ with the following two hyperplanes.

$$4x-3y+0.73z =$$ 0

$$-x+2.1y+7z = 0.2$$

1. Construct a start system for the polynomial continuation using phc -r.
2. Perform the continuation with this start sytem using phc -p.