Getting the error message: The dimension of the ideal is 1, but it should be 0

asked 2017-12-31 23:24:09 +0200

david_c
31 ●3 ●3 ●7

I'm trying to solve a system of 5 polynomial equations.

Here's the code I tried to run:

P.<b, qa, qb, qc, qd>=PolynomialRing(QQ,order='degrevlex')
eq1=P(-25997.02495*qc+73589.75314*qa+19275.89428*qb^3+42024.09724*qc^3-35275.79436*qd^3+31409.96375*qd-22475.53767*qb+11165.49567*qc*qa*qd+38392.81504*qd*qb*qc-5354.736466*qc*qa*qb-40769.13796*qd*qb*qa-50708.36034*qc^2*qa-67780.84581*qd*qb^2+39326.95066*qd^2*qb+5359.28038*qb^2*qa-35437.12402*qd*qc^2+48529.90789*qd^2*qc-6801.32966*qb^2*qc+13747.85604*qc^2*qb+9197.21841*qd^2*qa+2*b*qa)
eq2=-P(1938.516702*qc-9153.752714*qa-15279.24300*qb^3+8131.520743*qc^3+35208.27094*qd^3-25334.07110*qd+71321.57867*qb+38392.81500*qc*qa*qd-48554.44832*qd*qb*qc-64028.43710*qc*qa*qb-58696.44966*qd*qb*qa+2*b*qb+426.071103*qc^2*qa+26980.57949*qd*qb^2-29944.11326*qd^2*qb+4540.542985*qb^2*qa-1130.66934*qd*qc^2-8809.128406*qd^2*qc-9514.136166*qb^2*qc-21010.04302*qc^2*qb+26005.16567*qd^2*qa)
eq3=P(1938.516702*qc-9153.752714*qa-15279.24300*qb^3+8131.520743*qc^3+35208.27094*qd^3-25334.07110*qd+71321.57867*qb+38392.81500*qc*qa*qd-48554.44832*qd*qb*qc-64028.43710*qc*qa*qb-58696.44966*qd*qb*qa+2*b*qb+426.071103*qc^2*qa+26980.57949*qd*qb^2-29944.11326*qd^2*qb+4540.542985*qb^2*qa-1130.66934*qd*qc^2-8809.128406*qd^2*qc-9514.136166*qb^2*qc-21010.04302*qc^2*qb+26005.16567*qd^2*qa)
eq4=P(2123.141846*qc+5788.216407*qa+22583.23914*qb^3-8500.597767*qc^3-9489.15234*qd^3+75013.55778*qd-25334.07110*qb+46634.03800*qc*qa*qd-22972.99327*qd*qb*qc+38392.81504*qc*qa*qb+38688.54642*qd*qb*qa+2*b*qd-9815.37666*qc^2*qa-33782.05127*qd*qb^2+64855.67483*qd^2*qb-42159.09845*qb^2*qa-35072.07256*qd*qc^2-8694.617828*qd^2*qc-29859.97200*qb^2*qc-1130.66933*qc^2*qb-3340.393775*qd^2*qa)
eq5= P(qa^2+qb^2+qc^2+qd^2-1)
I = Ideal(eq1, eq2, eq3, eq4, eq5)
I.groebner_basis('libsingular:std')
I.variety(RR)

At the last line of code:

I.variety(RR)

I get this error:

The dimension of the ideal is 1, but it should be 0

Anyone knows why? or alternatively, how to get the solutions of the equations?

Thanks

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The codimension is not the number linearly independent equations in general. You have variety of dimension 2 in C^4 that can not be obtained with 2 equations. You can have a look at algebraic dimension (wikipedia).

vdelecroix ( 2018-01-01 12:11:05 +0200 )edit

I will look at it thanks. It was just guess. The reason I said that is that if I write: I = Ideal(eq1) I.dimension() I get 4. and if I define the ideal with 2 equations the dimension is 3 and so on.

david_c ( 2018-01-01 14:15:44 +0200 )edit

You are mostly right of course!

vdelecroix ( 2018-01-01 19:25:41 +0200 )edit

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