# Solve nonlinear system with eight variables

Please I realy need Help We consider the three functions H1=4(x+b1 y)^2+8(c1x-d1 y)+e1^2 y^2, with e1>0 H2=4(x+b2 y)^2+8(c2 x-d2 y)+e2^2 y^2, with e2>0 H3=4(x+b3 y)^2+8(c3 x-d3 y)+e3^2 y^2, with e3>0...d1,b1,c1,d2,c2,b2, d3,c3,b3 are real parameters. I would like to solve (to get the values of alpha, beta, gamma, delta, f,h,g,k) the following system or (if it is not possible) to give the maximal number of its solutions The system is E1=E2=E3=E4=E5=E6=E7=E8=0 such that E1=H1(alpha,beta)-H1(gamma,delta) , E2=H2(alpha,beta)-H2(f,g), E3=H2(gamma,delta)-H2(h,k), E4=H3(h,k)-H3(f,g), E5=beta^2-alpha(alpha-1)(alpha-3), E6=delta^2-gamma(gamma-1)(gamma-3), E7=g^2-f(f-1)(f-3), E8=k^2-h(h-1)(h-3).

• Please provide at least valid Mathematica code for the equations and example values for the parameters. Having to guess the parameter ranges for which you need solutions is quite a pain. Mar 16 '19 at 8:50
• Probably not feasible to get an analytic solution. For specified numeric values of the parameters it might be possible to get solutions using NSolve and also FindRoot should work for getting a single solution. As for counting them, if the need is to get a max count on real solutions subject to the parameter constraints, again that will not be feasible. Counting complex-valued solutions is a different matter-- just plug in random values for the parameters and if NSolve handles that, you have your result. Mar 16 '19 at 14:38
• Thank you sir for your attention of my question Mar 16 '19 at 15:48
• If you give the context for this problem, there may be other, more geometric methods to find solutions. Mar 16 '19 at 19:07
• What do you mean by context? Mar 16 '19 at 19:39

Translating your question to Mathematica to get you started:

H1[x_, y_] = 4 (x + b1 y)^2 + 8 (c1 x - d1 y) + e1^2 y^2;
H2[x_, y_] = 4 (x + b2 y)^2 + 8 (c2 x - d2 y) + e2^2 y^2;
H3[x_, y_] = 4 (x + b3 y)^2 + 8 (c3 x - d3 y) + e3^2 y^2;

E1 = H1[α, β] - H1[γ, δ];
E2 = H2[α, β] - H2[f, g];
E3 = H2[γ, δ] - H2[h, k];
E4 = H3[h, k] - H3[f, g];
E5 = β^2 - α (α - 1) (α - 3);
E6 = δ^2 - γ (γ - 1) (γ - 3);
E7 = g^2 - f (f - 1) (f - 3);
E8 = k^2 - h (h - 1) (h - 3);

Assuming[Element[{d1, b1, c1, d2, c2, b2, d3, c3, b3}, Reals] &&
e1 > 0 && e2 > 0 && e3 > 0,
Solve[E1 == E2 == E3 == E4 == E5 == E6 == E7 == E8 == 0,
{α, β, γ, δ, f, h, g, k}]]

• I did it but it is not easy and mathematica did not give ,e a result Mar 16 '19 at 10:45