Georgi Guninski on Mon, 12 May 2014 14:38:21 +0200

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Heuristic that might speed some thue equations with no solutions

I am not good at theory, but this appears plausible to me.

Suppose you want to solve 
thue(f(x),A) [1]
where $f(x)$ is irreducible.

Let $K$ be the number field with defining polynomial
$f(x)$. Every solution to [1] $u,v$ satisfies 
norm(v x- u)=A.

If $A$ is not an integer norm in $K$, [1] has no solution.
bnfisintnorm() appears significantly faster than thue().

Is it sound first to check bnfisintnorm() and
if it fails to not bother with thue(), assuming
f(x) is (monic) irreducible?


? a=20^30+3;K=bnfinit(x^4+1);a%4
%30 = 3
? bnfisintnorm(K,a)
%31 = [] \\ fast
? th=thueinit(K.pol);
? \g 3
   debug = 3
? thue(th,a)
* Checking for small solutions <= 8095430811