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Understanding sqrt of sage

asked 2020-12-19 14:38:37 +0200

anonymous user

Anonymous

updated 2020-12-19 15:22:13 +0200

I'm reading the following piece of code:

p = 10000000000000001119

R.<x> = GF(p)[]; y=x
f = y + prod(map(eval, 'yyyyyyy'))
C = HyperellipticCurve(f, 0)
J = C.jacobian()

test=[J(C(x, min(f(x).sqrt(0,1)))) for x in (11,22,33)]

where $J$ is the jacobian of my hyper elliptic curve $C$.

I don't undestad what the sqrt function with those two arguments does. Reading from the docs it seems that I'm enabling the all flag:

all - Whether to return a list of all square roots or just a square root (default: False)

Can you explain what is mathematically doing that sqrt?

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You should provide the construction of J, C and f so that people can reproduce your situation.

tmonteil gravatar imagetmonteil ( 2020-12-19 14:55:05 +0200 )edit

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answered 2020-12-19 16:10:09 +0200

slelievre gravatar image

In many settings,

  • the square function $x \mapsto x^2$ is mostly two-to-one, except at zero
  • given $y$ in some ring $R$ with a product operation, there can be zero, one or two values of $x$ for which $x^2 = y$ (i.e. $x \cdot x = y$). When two such $x$ exist, they are opposites of each other, and called "the square roots of $y$"
  • in many cases there is a way to extend the ring $R$ to a bigger ring $S$ where all nonzero elements have two square roots. For example if $R$ is the field $\mathbb{R}$ of real numbers then we can take $S$ to be the field $\mathbb{C}$ of complex numbers

When working with the real numbers, - zero is its own unique square root - positive numbers have two square roots in the real numbers - negative numbers have no square root in the real numbers, but two square roots in the complex numbers

The two square roots of a positive number being opposite to each other, one is negative and one is positive. Since we find positive numbers more "natural", we often decide to give the positive root a special role and to call it "square root of $y$", which could be thought of as short for "the positive square root of $y$". The other root is then referred to as "minus square root of $y$".

In computer algebra systems, one has to decide what to do when the user asks for square roots a command such as sqrt(y) or y.sqrt():

  • regarding where to look for square roots:
    • stay in the ring where $y$ lives
    • or extend it, if necessary, to a larger ring where square roots exist, if they did not in the ring where $y$ lives
  • regarding how many square roots to return:
    • return all the square roots of its argument,
    • or return one particular square root (e.g. the positive one when that makes sense)?

I suppose the documentation you are referring to is the SageMath documentation for sage.structure.element.CommutativeRingElement.sqrt.

It explains that y.sqrt() takes two optional parameters, extend to decide whether to look for square roots in a larger ring if necessary, and all to decide whether to return all square roots or a preferred square root when there is one. There's an additional optional parameter name to name the extra generator of the new ring if you decide to extend.

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Asked: 2020-12-19 14:38:37 +0200

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Last updated: Dec 19 '20