Charles Greathouse on Wed, 06 Apr 2011 19:42:14 +0200 |
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Re: Polynomial divisibility |
On Wed, Apr 06, 2011 at 11:56:40AM -0400, Charles Greathouse wrote:Assuming that p is prime and P has integral coefficients, why not use
> I wanted to know if there's an efficient way to count the number of
> residue classes (mod p) for which the polynomial is divisible by p.
> The straightforward approach
>
> sum(n=1,p,substpol(P,x,Mod(n,p))==0)
>
> is slow.
>
> In my case the polynomial is reducible and of degree 62 with
> 'reasonable' coefficients (wordsize on a 64-bit machine, the largest
> is 44 bits). I could test the smaller polynomials first but I think
> the overhead would be more expensive than the benefit -- it's rare
> that p will divide any given value of the polynomial.
polrootsmod ?
You can do a bit better though:
poldegree(FpX_gcd(P,FpXQ_pow(x,p,P,p)-x))
with the suitable install() invocation.
Cheers,
Bill.