phc-course

Reinhard Steffens (steffens@math.uni-frankfurt.de)
http://www.math.uni-frankfurt.de/~steffens/phc/

Lets start with a very easy example.

We want to solve the following system of equations.

$$x^2+y^2-1 =$$ 0

$$x^3+y^3-1 =$$ 0

1. Type

   phc -b
   

   in a command line.
2. phc asks you, if your system is on a file. Say no by typing 'n'.
3. Now you will be asked for the number of polynomials and the number of variables.
4. In the next step we have to enter the two polynomials above. (Don't type '=0').

   x^2+y^2-1;
   2*x^3+y^3-4;
   

5. Choose a name for an output file.
6. Read the output file with an editor of your choice.

The solutions are presented in the following form.

The output file contains much more than just the solutions to the above system. See http://www.math.uni-frankfurt.de/~steffens/phc/phc-intro/node7.html for a few more details.

So lets get slightly more advanced.

We learned in Jim Ruffo's delightful seminar talk that to find the lines meeting 4 given general lines we have to solve

$$\det \begin{pmatrix}1 & t & t^2 & t^3 0 & 1 & 2t & 3t^2 1 & 0 & a & b 0 & 1 & c & d \end{pmatrix} =0$$

for $t= -1,0,1,2$ . Jim said this system has 2 solutions, namely

$$a=\frac{1}{6}\pm \frac{\sqrt{13}}{6}, \quad b=c=1, \quad d=3a-1 .$$

So lets check if he did that right.

Open an editor of your choice and enter the system of polynomial equations we want to solve. The system is preceded by the number of equations and variables involved.

 4
a*d - b*c - 3*a - 2*b + 2*c + d + 1;
a*d - b*c;
a*d - b*c - 3*a + 2*b - 2*c + d + 1;
a*d - b*c - 12*a + 4*b - 16*c + 4*d + 16;

Save this file as `4lines.in'.

First we check that the number of solutions is correct. For this we use the mixed volume.

Type phc -m 4lines.in 4lines.mv to start. We want to use static lifting and we have no mixed subdivision and also we do not want to enforce a certain type of mixture. Now eliminate the non-vertex points and tell phc to choose random linear vectors for the lifting. We want to use floating point lifting values, we don't want the mixed cells on a seperate file and we don't want to perform polyhedral continuation (at least not now).

Check the output file '4lines.mv'. On a well hidden place you will find the line 'The mixed volume equals : 2'. If not you did something wrong.

Finally we want to compute the solutions.

Type phc -b 4lines.in 4lines.out to start the black-box solver. Read the output file and compare the solutions to the ones given above.

For more information on what phc does see my short introduction http://www.math.uni-frankfurt.de/~steffens/phc/phc-intro/index.html.

When you think you know enough, try to solve the questions in the exercise section http://www.math.uni-frankfurt.de/~steffens/phc/phc-intro/node17.html.

Enjoy.