# solving simultaneous equations trouble using solve()

Hi, I am having a lot of difficulty solving the following set of equations using solve. I am gonna insert my code below:

  sage:var('x1,x2,x3,w1,w2,w3')
sage:f=w1+w2+w3==2;g=(w1*x1)+(w2*x2)+(w3*x3)==0;h=(w1*(x1)^2)+(w2*(x2)^2)+(w3*(x3)^2)==(2/3);j=(w1*(x1)^3)+(w2*(x2)^3)+(w3*(x3)^3)==0;k=(w1*(x1)^4)+(w2*(x2)^4)+(w3*(x3)^4)==(2/5);l=(w1*(x1)^5)+(w2*(x2)^5)+(w3*(x3)^5)==0
sage: solve([f,g,h,j,k,l],w1,w2,w3,x1,x2,x3,solution_dict=True)


The output is:

 [{w1 + w2 + w3: 2},
{w1*x1 + w2*x2 + w3*x3: 0},
{w1*x1^2 + w2*x2^2 + w3*x3^2: 2/3},
{w1*x1^3 + w2*x2^3 + w3*x3^3: 0},
{w1*x1^4 + w2*x2^4 + w3*x3^4: 2/5},
{w1*x1^5 + w2*x2^5 + w3*x3^5: 0}]


obviously this output doesn't help me as I need the solution for all six variables. Any help is appreciated please.

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For systems of polynomial equations, do not use the symbolic ring, but ideals in polynomial rings. See https://doc.sagemath.org/html/en/refe...

( 2021-12-31 19:45:31 +0200 )edit

@FrédéricC I am not sure how to do so. Could you elaborate a little more ?

( 2021-12-31 20:09:06 +0200 )edit

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For all those interested I have manged to do what I wanted using sympy.

 sage: var('x1,x2,x3,w1,w2,w3')
sage: f=w1+w2+w3==2;g=(w1*x1)+(w2*x2)+(w3*x3)==0;h=(w1*(x1)^2)+(w2*(x2)^2)+(w3*(x3)^2)==(2/3);j=(w1*(x1)^3)+(w2*(x2)^3)+(w3*(x3)^3)==0;k=(w1*(x1)^4)+(w2*(x2)^4)+(w3*(x3)^4)==(2/5);l=(w1*(x1)^5)+(w2*(x2)^5)+(w3*(x3)^5)==0
sage: import sympy
sage sympy.solve([f,g,h,j,k,l],w1,w2,w3,x1,x2,x3)


output:

  [{w1: 5/9, w2: 5/9, w3: 8/9, x1: -sqrt(15)/5, x2: sqrt(15)/5, x3: 0},
{w1: 5/9, w2: 5/9, w3: 8/9, x1: sqrt(15)/5, x2: -sqrt(15)/5, x3: 0},
{w1: 5/9, w2: 8/9, w3: 5/9, x1: -sqrt(15)/5, x2: 0, x3: sqrt(15)/5},
{w1: 5/9, w2: 8/9, w3: 5/9, x1: sqrt(15)/5, x2: 0, x3: -sqrt(15)/5},
{w1: 8/9, w2: 5/9, w3: 5/9, x1: 0, x2: -sqrt(15)/5, x3: sqrt(15)/5},
{w1: 8/9, w2: 5/9, w3: 5/9, x1: 0, x2: sqrt(15)/5, x3: -sqrt(15)/5}]


If anyone can enlighten me on how to use the polynomial rings to solve polynomial systems of equations I'd be very happy (I am also a noob at sagemath btw)

Thank you.

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Here is a minimal illustrating example:

sage: R = PolynomialRing(QQbar,'x,y')
sage: x, y = R.gens()
sage: I = R.ideal([x*x+y*y-1, x-5*y])
sage: I.variety()
[{y: -0.1961161351381841?, x: -0.9805806756909202?},
{y: 0.1961161351381841?, x: 0.9805806756909202?}]

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How do I even use any of this ? 1) what is the R.gens() 2) what is the R.ideal ? 3) what is I.variety ? 4) isn't there a simpler way to solve polynomials ?

( 2021-12-31 22:17:03 +0200 )edit
1

The way suggested by FredericCis the simplest way to tackle general polynomial systems in Sage (and in general...).

See chapters 7 and 9 of this book for an introduction to the subject. Chapter 9 points to a textbook, which turns out to be easily Googleable,

BTW, I'd recommend reading the whole Sage book (and a good Python tutorial) to gain a deeper understanding of Sage...

( 2022-01-01 12:11:47 +0200 )edit