Post new topic Reply to topic  [ 5 posts ] 
Author Message
 Post subject: SICA p.9 Exercise 1.1.13.
PostPosted: Thu May 02, 2013 9:46 am 
In A Singular Introduction to Commutative Algebra, the exercise says: "Write a Singular procedure, depending on two integers p, d, with
p a prime, which returns all polynomials in F_p[x] of degree d such that the corresponding polynomial function vanishes. Use the procedure to display all f ∈ (Z/5Z)[x] of degree ≤ 6 such that f = 0."

My solution:

Code:
proc zeroFunctions (int p, int d)
{  list l; //list of polynomials of degree d whose function is zero everywhere
   for (int i=p^d; i<p^(d+1); i=i+1) //the decimals of i in the numeral system with basis p determine the coefficients of f
       { poly f=0; int j,e=i,0; //e is the exponent
         while(j>0){ f=f+(j%p)*x^e; e=e+1; j=j / p; }
         int b=1; //boolean expression 'true'
         for (int k=0; k<p; k=k+1)
             { b = b and subst(f,x,k)==0; } //b remains true if f(j)=0
         if (b==1) { l = l + list(f); } //if f is the zero function, we append it to l }
   return(l);}
int p=5;   ring R=p,(x),dp;   zeroFunctions(p,6);


This works correctly for p=2, d=3. I don't know exactly what happens for p=5, d=6. The computer seems to be stuck/frozen. Out of memory? Endless loop? Too long output?

Question: How can I fix my procedure to work flawlessly?
Question: How can I tell Singular not to print every message "redefining b" and "int division", i.e. I'd like to switch to 'silent mode'.


Report this post
Top
  
Reply with quote  
 Post subject: Re: SICA p.9 Exercise 1.1.13.
PostPosted: Thu May 02, 2013 1:50 pm 

Joined: Wed Mar 03, 2010 5:08 pm
Posts: 103
Location: Germany,
This works correctly for p=2, d=3. I don't know exactly what happens for p=5, d=6. The computer seems to be stuck/frozen. Out of memory? Endless loop? Too long output?

Question1: How can I fix my procedure to work flawlessly?
Question2: How can I tell Singular not to print every message "redefining b" and "int division", i.e. I'd like to switch to 'silent mode'.[/quote]

ad Question2; 1. Singular tells you that you should write
Code:
j=j div p;

instead of j = j / p;
2. You could surpress the "redefine messages" by setting
Code:
option(noredefine);

The reason that these messages occur, is that you define local
variables inside the nested loops. So better you define at least
Code:
int j,e,b,k;
poly f;

at the very beginning and only set them in the loops.
The first for loop you may leave with the declaration as it is
Code:
for (int i=  )
{

}

General remark: Your list will have duplicate entries!


Report this post
Top
 Profile  
Reply with quote  
 Post subject: Re: SICA p.9 Exercise 1.1.13.
PostPosted: Sat May 04, 2013 3:27 pm 
How can there be duplications in the list l? Every number j uniquely determines the coefficients of the polynomial f. Here p is assumed to be a prime number.


Report this post
Top
  
Reply with quote  
 Post subject: Re: SICA p.9 Exercise 1.1.13.
PostPosted: Mon May 06, 2013 2:39 pm 

Joined: Wed Mar 03, 2010 5:08 pm
Posts: 103
Location: Germany,
OK; I don't remember what lead me to that apparently false claim.

But I can't see why your code should not work. If I call zeroFunctions (5,6)
in char 5, it gives immediately a list with 54 entries.


Report this post
Top
 Profile  
Reply with quote  
 Post subject: Re: SICA p.9 Exercise 1.1.13.
PostPosted: Tue May 07, 2013 2:31 pm 
If I type
Code:
proc zeroFunctions (int p, int d)
{ option(noredefine); //no unnecessary "redefine f,j,e,k" notifications
list l; //list of polynomials
for (int i=p^d; i<p^(d+1); i=i+1)
{ poly f=0; int j,e=i,0; //e is the exponent
while(j>0){ f=f+(j%p)*x^e; e=e+1; j=j div p; }
int b=1; //boolean expression ’true’
for (int k=0; k<p; k=k+1){ b = b and subst(f,x,k)==0; } //b remains true iff f(k)==0
if (b==1) { l=l+list(f); } } //if f is the zero function, we append it to l
return(l); }
int p,d=5,6; ring R=p,(x),dp; zeroFunctions(p,d);

then Singular returns 20 polynomials, which are all nonzero of degree 6. I don't think there are any duplications.

I suspect that you didn't set the ring to ring R=p,(x),dp;.

Thank you for your help!
regards, Leon


Report this post
Top
  
Reply with quote  
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 5 posts ] 

You can post new topics in this forum
You can reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

It is currently Mon Dec 18, 2017 3:08 am
cron
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group