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

["Thomas D. Dean" <tomdean@wavecable.com>]

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.

? P=Pol([5, -22, 11, 19, -7])
%7 = 5*x^4 - 22*x^3 + 11*x^2 + 19*x - 7
? polroots(P)
%8 = [-0.83424318431392171711562612580235620408 + 0.E-38*I, 
0.34337956895288963385776743154236596799 + 0.E-38*I, 
1.4000000000000000000000000000000000000 + 0.E-38*I, 
3.4908636153610320832578586942599902361 + 0.E-38*I]~
? polsturm(P)
%9 = 4
? polsturm(P-37*x+x^2)
%10 = 2
? polsturm(P-37*x+x^2+445)
%11 = 0
? #polsturm(P-37*x+x^2+445)
%12 = 0
? #polsturm(P)
%13 = 1

Tom Dean