# System of 15 equations in 23 variables

Discussion in 'Mathematics' started by LCH, Aug 2, 2020 at 2:25 AM.

1. ### LCHGuest

$$1-k_0^2=0, \frac{1}{2} \left(-7 \alpha ^2+38 \alpha +2 c_3 (\alpha +\beta-3)+2 c_6-2 k_1^2-2 k_5^2-48\right)=0, c_5-k_2^2-k_{10}^2=0, c_8(\alpha +\beta -5)+5 c_{10}-k_3^2-k_6^2-k_8^2-k_{11}^2-2 k_2 k_4=0, c_{11}(\alpha +\beta -7)-k_4^2-k_7^2-k_9^2-k_{12}^2=0, -2 \alpha +\beta +2 c_3-2k_0 k_1+4=0, c_2-2 k_0 k_2=0, c_6-2 k_0 k_3=0, c_9-2 k_0 k_4=0,-\frac{3}{2} (\alpha -2) (3 \alpha -11)+c_2 (\alpha +\beta -3)+3 c_5+2 c_6-2 k_1 k_2=0, (4-\alpha ) (\alpha -2) (2 \alpha -9)+c_6 (\alpha +\beta -4)+2c_8+4 c_9-2 k_1 k_3-2 k_5 k_6=0, -\frac{1}{8} (\alpha -6) (\alpha -5)(\alpha -4) (\alpha -2)+c_9 (\alpha +\beta -5)+c_{10}-2 k_1 k_4-2 k_5k_7=0, c_5 (\alpha +\beta -4)+3 c_8-2 k_2 k_3=0, c_8 (\alpha +\beta -5)+5 c_{10}-k_3^2-k_6^2-k_8^2-k_{11}^2-2 k_2 k_4=0, c_{10} (\alpha +\beta-6)+7 c_{11}-2 \left(k_3 k_4+k_6 k_7+k_8 k_9\right)=0$$

I use FindInstance to find a set of real parameters (here $\alpha$, $\beta>0$ and other parameters are real) which satisfy the above 15 equations. Perhaps there are lots of parameters, I didn't get any answer from Mathematica. Any reference, suggestion, idea, alternative solution or comment is welcome. Thank you! (the code I used is as follows.)

FindInstance[10 - 5*\[Alpha] + 2*Subscript[c, 2] + Subscript[c, 3]==0 && 1 - Subscript[k, 0]^2 == 0 && (1/2)*(-48 + 38*\[Alpha] - 7*\[Alpha]^2 + 2*(-3 + \[Alpha] + \[Beta])*Subscript[c, 3] + 2*Subscript[c, 6] - 2*Subscript[k, 1]^2 - 2*Subscript[k, 5]^2) == 0 && Subscript[c, 5] - Subscript[k, 2]^2 - Subscript[k, 10]^2 == 0 && (-5 + \[Alpha] + \[Beta])*Subscript[c, 8] + 5*Subscript[c, 10] - Subscript[k, 3]^2 - 2*Subscript[k, 2]*Subscript[k, 4] - Subscript[k, 6]^2 - Subscript[k, 8]^2 - Subscript[k, 11]^2 == 0 && (-7 + \[Alpha] + \[Beta])*Subscript[c, 11] - Subscript[k, 4]^2 - Subscript[k, 7]^2 - Subscript[k, 9]^2 - Subscript[k, 12]^2 == 0 && 4 - 2*\[Alpha] + \[Beta] + 2*Subscript[c, 3] - 2*Subscript[k, 0]*Subscript[k, 1] == 0 && Subscript[c, 2] - 2*Subscript[k, 0]*Subscript[k, 2] == 0 && Subscript[c, 6] - 2*Subscript[k, 0]*Subscript[k, 3] == 0 && Subscript[c, 9] - 2*Subscript[k, 0]*Subscript[k, 4] == 0 && (-(3/2))*(-2 + \[Alpha])*(-11 + 3*\[Alpha]) + (-3 + \[Alpha] + \[Beta])*Subscript[c, 2] + 3*Subscript[c, 5] + 2*Subscript[c, 6] - 2*Subscript[k, 1]*Subscript[k, 2] == 0 && (-(-4 + \[Alpha]))*(-2 + \[Alpha])*(-9 + 2*\[Alpha]) + (-4 + \[Alpha] + \[Beta])*Subscript[c, 6] + 2*Subscript[c, 8] + 4*Subscript[c, 9] -2*Subscript[k, 1]*Subscript[k, 3] - 2*Subscript[k, 5]*Subscript[k, 6] == 0 && (-(1/8))*(-6 + \[Alpha])*(-5 + \[Alpha])*(-4 + \[Alpha])*(-2 + \[Alpha]) + (-5 + \[Alpha] + \[Beta])*Subscript[c, 9] + Subscript[c, 10] - 2*Subscript[k, 1]*Subscript[k, 4] - 2*Subscript[k, 5]*Subscript[k, 7] == 0 && (-4 + \[Alpha] + \[Beta])*Subscript[c, 5] + 3*Subscript[c, 8] - 2*Subscript[k, 2]*Subscript[k, 3] == 0 && (-5 + \[Alpha] + \[Beta])*Subscript[c, 8] + 5*Subscript[c, 10] - Subscript[k, 3]^2 - 2*Subscript[k, 2]*Subscript[k, 4] - Subscript[k, 6]^2 - Subscript[k, 8]^2 - Subscript[k, 11]^2 == 0 && (-6 + \[Alpha] + \[Beta])*Subscript[c, 10] + 7*Subscript[c, 11] - 2*(Subscript[k, 3]*Subscript[k, 4] + Subscript[k, 6]*Subscript[k, 7] + Subscript[k, 8]*Subscript[k, 9]) == 0, {\[Alpha], \[Beta], Subscript[k, 0], Subscript[k, 1], Subscript[k, 2], Subscript[k, 3], Subscript[k, 4], Subscript[k, 5], Subscript[k, 6], Subscript[k, 7], Subscript[k, 8], Subscript[k, 9], Subscript[k, 10], Subscript[k, 11], Subscript[k, 12], Subscript[c, 2], Subscript[c, 3], Subscript[c, 5], Subscript[c, 6], Subscript[c, 8], Subscript[c, 9], Subscript[c, 10], Subscript[c, 11]}, Reals]