John Cremona on Mon, 27 Nov 2023 09:30:51 +0100


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Re: asking for a simple locally soluble algorithm for a quartic

I think (given the context) that Randall wants local solubility of f(x)=y^2, not f(x)=0.  I think Bill must have implemented this in pari when he implemented full 2-descent (as used in ellrank()) so the question is then about making an internal pari function available via gp.

John

On Mon, 27 Nov 2023 at 07:39, Thomas D. Dean <tomdean@wavecable.com> wrote:
On 11/26/23 23:22, Thomas D. Dean wrote:
> On 11/26/23 17:44, American Citizen wrote:
>> Does anyone have a simple GP-Pari script which outputs 0 for false and
>> 1 for true when the input is a quartic in vector format: [a,b,c,d,e]
>> where the quartic is a*x^4 + b*x^3 + c^x^2 + d^x +e and we are trying
>> to find the everywhere_local_solubility of the quartic?
>>
>> Example GP Pari function:
>>
>> everywhere_locally_soluble(Q) = [0,1] where 0 means false and 1 means
>> true.
>>
>>
> look at polsturm

for real roots

Sorry, I hit send too soon

Some hints in:
https://en.wikipedia.org/wiki/Quartic_equation

Tom Dean