On Tue, Jul 04, 2023 at 10:48:10PM +0200, Ralf Hemmecke wrote:
> (458) -> sqrt(s2)
>
> +--------------------------------------------------------+
> | +--+
> \|1097675101059710976000 %i\|67 + 36223278334970462208000
> (458) -----------------------------------------------------------
> 17
> Type: Expression(Complex(Integer))
>
> (459) -> factor gcd(1097675101059710976000,36223278334970462208000)
>
> 18 5 3 3 2
> (459) 2 3 5 11 10177
>
> Is there some easy way (a fricas function) to obtain from sqrt(s2) an
> expression that has moved all common quadratic parts under the square root
> outside?
>
> I would like to see something like
>
> +------------------+
> | +--+
> 2579258880 \|165 %i\|67 + 5445
> (471) --------------------------------
> 17
> Type: Expression(Complex(Integer))
>
Well, this is probably too much:
rootFactor(sqrt(p1))
+-------------+
+--------+ +------+ +-+ +--+ | +--+
(70) 2579258880 \|1 + 2 %i \|2 + %i \|3 \|11 \|\|67 - 33 %i
We have ingredients like
nthRoot(squareFree(1097675101059710976000), 2)$FactoredFunctions(Integer)
(61) [exponent = 2, coef = 2579258880, radicand = [3, 5, 11]]
qroot(1097675101059710976000/1, 2)$pR
(63) [exponent = 2, coef = 2579258880, radicand = 165]
and content and primitivePart:
p1 := 1097675101059710976000*%i*sqrt(67) + _
36223278334970462208000::EXPR(COMPLEX(INT))
content(numer(p1))
(65) 1097675101059710976000
primitivePart(numer(p1))
+--+
(66) \|67 - 33 %i
In a sense closest to what you want is 'froot' from PolynomialRoots
but it does not handle integer content (which AFAICS is delibarate
decision due to potentially prohibitive cost of integer factorization).
It would be easy to add variant of 'froot' which handles also
integer content. OTOH calling anything from PolynomialRoots
is far from easy. For example pR above was defined via this
sequence of assignments:
vV := Kernel(Expression(Complex(Integer)))
eE := IndexedExponents(vV)
sP := SparseMultivariatePolynomial(Complex(Integer), _
Kernel(Expression(Complex(Integer))))
pR := POLYROOT(eE, vV, Complex(Integer), sP, Expression(Complex(Integer)))
Probably better would be do such transformation in 'rootSimp',
but again they can be quite expensive, so debatable is should
be done by default. And of course, 'rootSimp' can do things
which you do not want...
BTW: For roots of rational numbers AlgebraicFunction is doing this
by defualt:
sqrt(1097675101059710976000)
+---+
(67) 2579258880 \|165
But such transformations sometimes are undesirable.
--
Waldek Hebisch
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