How to solve underdetermined systems of polynomial equations?

Question

I am trying to solve under determined simultaneous non - linear equations, where the variables are multiplied, but the power of the variables is always 1, is there a formal way doing it?

For eg: $x+y+z=6$, $xyz=6$

I was previously using reduced row Echelon form, but now that is less useful, is there a counterpart of Echelon matrix form for the above kind of equations?

Answer 1

When you have too many unknowns, an option is to consider some of them as free parameters and solve for the remaining ones. In the given example, solve

$$\begin{cases}x+y+z=6,\\xyz=6\end{cases}$$ where $z$ is assumed to be known.

We can eliminate $y$ by multiplying the first equation by $xz$,

$$\begin{cases}x^2z+xyz+xz^2=6xz,\\xyz=6\end{cases}$$

which leaves us

$$x^2z+x(z^2-6z)+6=0,$$ quadratic in $x$.

This simple trick turns an indeterminate system into a determinate one, but the real problem of solving the determinate one remains.

Answer 2

Hint: With $$z=\frac{6}{xy}$$ we get $$x^2y+xy^2-6xy+6=0$$ or $$y^2+y(6+x)+\frac{6}{x}=0$$ so we get $$y_{1,2}=-\frac{6+x}{2}\pm\sqrt{\left(\frac{6+x}{2}\right)^2-\frac{6}{x}}$$ The solution is a curve. And a Groebner bases is given by $$6-6yz+y^2z+yz^2,x+y+z-6$$