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Given the formula, I want to convert it to a Poly object and minimize the size of the generator. For example, if my formula is x^2+x+sqrt(x), I would expect to have only sqrt(x) in the generator. But in most of my attempts, the generator contains both x and sqrt(x). 

from sympy import *
print('first attempt:',Poly('x^2+x+sqrt(x)'))
x=Symbol('x', positive=True)
print('second attempt:',Poly(x**2+x+sqrt(x)))
print('third attempt:',str(Poly({1:1,2:1,4:1},gens=S('sqrt(x)'))))

first attempt: Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')
second attempt: Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')
third attempt: Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')

Only the third method was successful, but the code for it is ugly and hard to automatize (I want to convert it because I want to easily extract coefficients and powers, so it gives me nothing). Is there a nicer way to do it?

EDIT

Based on smichr response, I've created a function to reduce the size of the generator.

def _powr(formula):
    if formula.func==Pow:
        return formula.args
    else:
        return [formula,S('1')]
def reducegens(formula):
    pol=Poly(formula)
    newgens={}
    ind={}
    for gen in pol.gens:
        base,pw=_powr(gen)
        coef,_=pw.as_coeff_mul()
        ml=pw/coef
        if base**ml in newgens:
            newgens[base**ml]=gcd(newgens[base**ml],coef)
        else:
            newgens[base**ml]=coef
            ind[base**ml]=S('tmp'+str(len(ind)))
    for gen in pol.gens:
        base,pw=_powr(gen)
        coef,_=pw.as_coeff_mul()
        ml=pw/coef
        pol=pol.replace(gen,ind[base**ml]**(coef/newgens[base**ml]))
    newpol=Poly(pol.as_expr())
    for gen in newgens:
        newpol=newpol.replace(ind[gen],gen**newgens[gen])
    return newpol

I've tried it on several examples and it's not bad, but there is room for improvements.

print(reducegens(S('x^2+x+sqrt(x)')))
print(reducegens(S('x**2+y**2+x**(1/2)+x**(1/3)')))
print(reducegens(S('sqrt(x)+sqrt(z)+sqrt(x)*sqrt(z)')))
print(reducegens(S('sqrt(2)+sqrt(3)+sqrt(6)')))

Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')
Poly((x**(1/6))**12 + (x**(1/6))**3 + (x**(1/6))**2 + y**2, x**(1/6), y, domain='ZZ')
Poly((sqrt(x))*(sqrt(z)) + (sqrt(x)) + (sqrt(z)), sqrt(x), sqrt(z), domain='ZZ')
Poly((sqrt(2)) + (sqrt(3)) + (sqrt(6)), sqrt(2), sqrt(3), sqrt(6), domain='ZZ')

There first three examples are OK, but it would be nice to represent sqrt(6) as sqrt(2)*sqrt(3) and not as another generator's element.

urojony
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1 Answers1

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You can just replace sqrt(x) with y or (using symbol-trickery) replace it with a Symbol named sqrt(y) (or sqrt(y) which I do to show that it worked):

>>> from symyp.abc import x, y
>>> from sympy import Symbol
>>> (x**2+x+sqrt(x)).subs(sqrt(x), y)
y**4 + y**2 + y
>>> (x**2+x+sqrt(x)).subs(sqrt(x), Symbol('sqrt(y)'))
sqrt(y)**4 + sqrt(y)**2 + sqrt(y)

Either form now has a single symbol and will create a Poly with a single generator.

smichr
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  • Actually I thought about this. My first thought was that it will be quite hard to automatize, too. Now I see that I just need to calculate LCM of denominators of powers. But maybe there is a magical built-in solution. – urojony Apr 24 '20 at 16:59
  • There is `unrad` from solvers.py but that won't show the change of variables in this case -- but getting something other than None will tell you that such a generator exists and was unified by `unrad`. – smichr Apr 24 '20 at 17:07