Explicit solution to set of two nonlinear equations

Question

Is it possible to derive the solution of the following set of two nonlinear equations explicitly?

\begin{cases} Axy - x³y² + B = 0\\ Cxy - Dx²y³ + E = 0 \end{cases}

If not, how can I derive the influence of one of the parameters, say A, on the value of x and y? Is there any workaround to this? (approximation?)

Answer 1

Yes, the solutions can be determined by the use of Groebner bases. However, one has to be careful what you mean by "explicitly". For $C=D=0$ and $E\neq 0$ there is no solution of course, for example. How do you include this in a general formula ?
In general, the idea is to express, say, $x$ as a polynomial function of $y,A,B,C,D,E$. To see what this means, let me consider a special example, e.g., $B=0$, $C=D=E=1$. Then $$ x=A(Ay^3 - y - 1), $$ and $y$ and $A$ have to satisfy the polynomial relation $$ A^2y^4 - 2Ay^2 - Ay + 1=0, $$ which is a quadratic equation in $A$. So $y\neq 0$ is arbitrary, and $A$ and $x$ are determined by these formulae.

Answer 2

Elimination of variables is useful: If you have two equations $F(x,y)=G(x,y)=0$, it is possible in principle to get another equation $\phi(x)=0$ as a consequence of these. If the equations were linear you simply get $y$ in terms of $x$ from the first and plug into the second, but for higher degrees something else needs to be done. The new equation depends only on the $F$ and $G$ and in some cases can be obtained explicitly, for instance using Groebner bases.