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compute regulator with more precision

asked 2021-12-27 03:10:10 +0200

beeson gravatar image

updated 2022-01-22 06:07:28 +0200

I want to compute the regulator of a real quadratic field Q(sqrt d) to high precision, accurately enough to compute the fundamental unit. The default breaks at d = 331 where fundamental unit needs more than 53 bits (the precision of doubles). The documentation says that Pari computes to a higher precision than SageMath. Also somewhere it says that if you get a good enough approximation to the regulator, it's trivial to refine it to high accuracy. It refers to "the tutorial" without a link; I read the Pari-GP tutorials on algebraic number theory without finding any explanation of that remark. So actually there are two questions here: point me to an explanation of refining the computation of the regulator, and secondly, fix the following code so that it doesn't print "oops" when d = 331. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

gp.set_real_precision(256)  # doesn't seem to do anything

def check_unit(N):
        for d in range(10,N):
                if not is_squarefree(d):
                        continue
                K.<a> = QuadraticField(d)        
                G = K.unit_group()
                [x,y] = G.gen(1).value()
                x = abs(x) 
                R = K.regulator(None)
                twox = round(exp(R))
                x2 = twox/2
                y2 = round(twox/sqrt(d))/2
                print(d,x,x2,y,y2,exp(R)/2)
                if x != x2 or y != y2:
                        print("oops!")
                        return 
                if norm_is_negative(x,d):
                       print("norm is negative")

A shorter "minimal non-working example":

sage: d = 2352721672669                                                         
sage: K.<a> = QuadraticField(d)                                                 
sage: gp.default("realprecision",256)                                           
0
sage: R = K.regulator()

followed by many error messages.

Later: well, precision 512 seems to work. So I guess it's just that 256 isn't enough!

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Comments

to change real precision in gp run gp.default("realprecision",256)

Max Alekseyev gravatar imageMax Alekseyev ( 2021-12-27 19:31:24 +0200 )edit

Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

Use the 101010 button to insert the text of your code ; alternatively, select your code and use Ctrl-K : in both case, your code willbe isolated by newlines and indented with four spaces, which is interpreted as "respect newlines and indentation".

HTH,

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2021-12-27 22:58:52 +0200 )edit
slelievre gravatar imageslelievre ( 2021-12-28 07:12:38 +0200 )edit

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answered 2021-12-27 19:05:47 +0200

beeson gravatar image

Someone pointed me to Maurer's dissertation:

https://tuprints.ulb.tu-darmstadt.de/87/

which answers the theoretical part of my question. But that still leaves the second part unanswered.

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Comments

Doesn't Max Alexeyev's comment solve the problem ?

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2021-12-27 23:01:46 +0200 )edit

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Asked: 2021-12-27 03:10:10 +0200

Seen: 62 times

Last updated: Jan 22