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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?)

user185050
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2 Answers2

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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.

Dietrich Burde
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  • If I understand you correct, I can solve for x and y keeping one parameter, say A, as a variable, when I insert the values of the other parameters? (B, C, D and E, which are all different fom zero) My general question was: can I express x and y as a function of A,B,C,D and E? (which are all different from zero) – user185050 Jan 14 '15 at 11:39
  • Could you be more specific in how you derive your expressions? – user185050 Jan 14 '15 at 11:39
  • You can express $x$ as a function of $y,A,B,C,D,E$ via case distinction, as demonstrated. By the way, for $E\neq 0$ we have $y\neq 0$. – Dietrich Burde Jan 14 '15 at 12:16
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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.

orangeskid
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