On Tue, May 14, 2024 at 03:04:15PM +0200, Ralf Hemmecke wrote:
>
> In fact, the value xx is one of radicalRoots(pp) where
>
> pp :=
> x^4-2729960418308000*x^3-395258439243352250000*x^2-55499520947716391500000000*x-345363656226658026765625000000
>
> Interestingly, when I put xx into Mathematica, I get a much nicer
> expressions.
>
> In[15]:= p1 = Root[pp, 1] // ToRadicals
>
> Out[15]= 250 (2729960418308 + 1930373524352 Sqrt[2] -
> 23569 Sqrt[2 (13416226688183641 + 9486704869150589 Sqrt[2])])
>
> In[25]:= p3 = Root[pp, 3] // ToRadicals
>
> Out[25]= 250 (2729960418308 - 1930373524352 Sqrt[2] -
> 23569 I Sqrt[2 (-13416226688183641 + 9486704869150589 Sqrt[2])])
>
> Can I somehow "convince" FriCAS to return similarly "simple" radical
> expresssions?
With the attached patch I get:
(2) -> radical_solve(univariate(pp))
(2)
[
+--------------------------------------------------------------------+
| +-+
\|658730414260118770166403625000 \|2 + 931585485794307216540975125000
+
+-+
482593381088000 \|2 + 682490104577000
,
+--------------------------------------------------------------------+
| +-+
- \|658730414260118770166403625000 \|2 + 931585485794307216540975125000
+
+-+
482593381088000 \|2 + 682490104577000
,
+---+
\|- 1
*
+--------------------------------------------------------------------+
| +-+
\|658730414260118770166403625000 \|2 - 931585485794307216540975125000
+
+-+
- 482593381088000 \|2 + 682490104577000
,
-
+---+
\|- 1
*
ROOT
+-+
658730414260118770166403625000 \|2
+
- 931585485794307216540975125000
+
+-+
- 482593381088000 \|2 + 682490104577000
]
Type: Union(List(Expression(Integer)),...)
The result looks more complicated, but this is because FriCAS does
not pull squares outside square roots.
I would be easy to hook this into 'radicalSolve' so that such result
is obtainde by default.
--
Waldek Hebisch
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--- ../fr-build123/src/algebra/ROOTUT.spad 2024-05-14 21:03:24.609019133 +0000
+++ ROOTUT.spad 2024-05-16 13:04:03.656821580 +0000
@@ -14,6 +14,10 @@
++ complex_roots(p) computes roots of p in terms of radicals
++ separating real and complex roots.
++ Returns "failed" when unsuccessful.
+ radical_solve : UP -> Union(List(F), "failed")
+ ++ radical_solve(p) tries to solve polynomial p in terms
+ ++ of radicals.
+ ++ Returns "failed" when unsuccessful.
quartic2 : (UP, F) -> Union(r_rec, "failed")
++ quartic2(p) should be local but conditional
my_sqrt : F -> F
@@ -33,6 +37,8 @@
root_pair(a : F, b : F) : C_rec == [a, b]
+ my_imag := sqrt(-1)
+
root4(a : F) : F ==
rec := froot(a, 4)$PolynomialRoots(IndexedExponents(K), K, R, P, F)
p1 := monomial(1, rec.exponent)$UP - rec.radicand::UP
@@ -249,4 +254,14 @@
(d = 4) => quartic(p)
"failed"
+ radical_solve(p : UP) : Union(List(F), "failed") ==
+ r1u := complex_roots(p)
+ r1u case "failed" => "failed"
+ r1 := r1u@r_rec
+ res := reverse!(r1.reals)
+ for c1 in r1.complexes repeat
+ res := cons(c1.real + my_imag*c1.imag, res)
+ res := cons(c1.real - my_imag*c1.imag, res)
+ reverse!(res)
+
