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Question: Wrong minimal polynomials?

Question:Wrong minimal polynomials?

sursumCorda 1099 Maple 2024
algebraic-numbers minpoly
April 25 2024
1

Consider the following exact algebraic number expr: 
 

> 

restart;
> 

kernelopts('version'):
> 

Physics:-Version():
> 

expr := Physics:-`*`(2-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2), sqrt(1-(1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2))^2))+Physics:-`*`(Physics:-`*`(3+Physics:-`*`(4, 1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2)), 1/12), sqrt(3-Physics:-`*`(4, (1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2))^2)))+Physics:-`*`(Physics:-`*`(2, Physics:-`*`(2-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2), 1/3)), sqrt(Physics:-`*`(Physics:-`*`(2, 2-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2)), 1-Physics:-`*`(2, 1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2)))))+Physics:-`*`(Physics:-`*`(1+Physics:-`*`(2, 1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2)), 1/6), sqrt(Physics:-`*`(2, (1/4)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2)^2-Physics:-`*`(3, (1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345, index = 2)-1/2)^2))^2)))):
> 

CodeTools:-Usage(PolynomialTools:-MinimalPolynomial(expr));

memory used=0.97MiB, alloc change=12.00KiB, cpu time=31.00ms, real time=27.00ms, gc time=0ns

  

27*_X^2+70*_X-433

 (1)
> 

evalf[30](eval(27*_X^2+70*_X-433, _X = expr));

0.262508598220109180966e-6

 (2)
> 

"evalf((mp:=CodeTools:-Usage(evala(':-Minpoly'(expr,_X)))));"

memory used=70.25GiB, alloc change=48.00MiB, cpu time=23.63m, real time=21.81m, gc time=3.18m

  

-422.3889573*_X+2208.949800*_X^4-225.8139518*_X^3-230.2173498*_X^2+59.33811620+_X^10-19.92850932*_X^9+166.6440365*_X^8-755.9985571*_X^7+1997.484781*_X^6-3014.379082*_X^5

 (3)
> 

type(mp, polynom(rational, _X))

false

 (4)


 

Download wrong_minpoly.mw 
I would like to find its minimal polynomial (without a priori knowledge). 

According to the documentation, 

if expr is an exact algebraic number, and n and acc are not given, then PolynomialTools:-MinimalPolynomial(expr) will call `evala/Minpoly` to compute an exact minimal polynomial of expr. If a name is not specified for the variable x, then _X will be used.

Regretfully, it is easy to see that the minimal polynomial of expr cannot be the returned 27*_X^2 + 70*_X - 433. And when I invoke evala@Minpoly directly, the result is still not correct (and this evaluation takes a rather long time). 
Another help page mentions that: 

the call mp := evala(Minpoly(expr, _X)) computes the monic minimal polynomial of a in the variable _X over the field of rational numbers (or multivariate rational functions); the resulting polynomial will not contain any algebraic numbers or functions.

However, as type(mp, polynom(rational, _X)) gives false, we know that mp cannot be the desired minimal polynomial of expr either. 
So, what is the proper way to compute the minimal polynomial of expr in Maple? 

Code: 

use alpha=1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345,index=2)-1/2)**2) in 
	expr:=(1+alpha)*sqrt(1-alpha**2)+(3+4*alpha)/12*sqrt(3-4*alpha**2)+2*(1+alpha)/3*sqrt(2*(1+alpha)*(1-2*alpha))+(1+2*alpha)/6*sqrt(2*((1-alpha)**2-3*alpha**2)) 
end:
CodeTools:-Usage(PolynomialTools:-MinimalPolynomial(expr));

Of note, the minimal polynomial of an algebraic number zeta is the unique irreducible monic polynomial of smallest degree p(x) with rational coefficients such that p(zeta)=0 and whose leading coefficient is 1. 
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