Dear all,
How can we obtain the analytical solution (in the complex n-space C^n) to the following 2nd order system of polynomial equation with n variables?
a_{11}x_1x_1+a_{12}x_1x_2+...+a_{1n}x_1x_n=b_1
a_{21}x_2x_1+a_{22}x_2x_2+...+a_{2n}x_2x_n=b_2
.
a_{k1}x_kx_1+a_{k2}x_kx_2+...+a_{kn}x_kx_n=b_k
.
a_{n1}x_nx_1+a_{n2}x_nx_2+...+a_{nn}x_nx_n=b_n
where a_{ij} and b_k, i,j,k=1,2,...n are constant complex values, and x_i, i=1,2,...n are unknowns.
Kind regards,
Chao
Have you tried to use solve function in Matlab (using symbolic variables)?
Hi Chao,
I think your problem can be formulated as the following:
http://math.stackexchange.com/questions/623428/solve-for-x-in-y-xtax
Hi Chao,
It is if you formulate it as X^T*A*X=B where X = (x_1,x_2, ... , x_n) and B=(b1,b_2,...,b_n).
When the matrix A equals a diagonal matrix D, one has a system of independent quadratic equations for D_{i,i}*x_i^2-b_i. This shows that there are no solutions when D_{i,i}=0, but b_i \ne 0 for some i. When D_{i,i}=b_i=0, the variable x_i can take any value.
Hence, one cannot expect a single formula for the solutions (unless one assumes that A and b are "generic", e.g, the corresponding variables can be treated as algebraic independent elements of a function field).
Dear Chao, thanks for the interesting question.
For a small number of equations (n = 2, 3) the analytical solution can be found using the Groebner bases. For example, for n = 2 we derive the resultant
a22*(a11*a22-a12*a21)*y^4-(2*a11*a22*b2-a12*a21*b2+a21^2*b1)*y^2+a11*b2^2,
which is the biquadratic expression in y. Its roots are the y-components of the solution. Similarly for n =3 we get the biquartic resultant, and its roots can always be expressed analytically in terms of a_{ij}, b_i.
I think that for n > 3 there is no a closed-form solution to your system. But you can still use the Groebner bases to efficiently derive the roots as follows. Construct the Groebner basis w.r.t. the grevlex order x_1 > ... > x_n. Then find the corresponding action matrix of size 2^n x 2^n. Its eigenvectors encode all 2^n roots of your system. The details can be found in the textbook by Cox, Little, O'Shea.
You have n^2 + n parameters. Unless n is rather small, this is a hopelessly large problem.
The Dixon resultant is far superior to Groebner bases when there are parameters, but your system is very generic. Google "Dixon resultant".
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