# Finding the Inverse of Polynomial Equations (Approximatly)

Assume one is given a set of two equations of the form: $$x(u,v) = u + a_1 u^2 + b_1 u v + c_1 v^2$$ $$y(u,v) = v + a_2 u^2 + b_2 u v + c_2 v^2$$ And one would like to find the inverse functions, i.e., $u(x,y), v(x,y)$. Clearly, the full solution is very very complex.

But what if we know that: $$a_i, b_i, c_i \ll u,v \ll 1$$?

Then approximately, we should be able to write the inverse formula using the same form: $$u(x,y) \approx x + a'_1 x^2 + b'_1 x y + c'_1 y^2$$ $$v(x,y) \approx y + a'_2 x^2 + b'_2 x y + c'_2 y^2$$

The question then becomes:

In the limit of small $a_i, b_i, c_i$, how can we express $a'_i, b'_i, c'_i$ in terms of $a_i, b_i, c_i$?

• Why is this approximation? Formula really becomes cumbersome, but it is for such a system is no problem. You can even decide the case of the system of Diophantine equations such as as there. artofproblemsolving.com/community/c3046h1048039__10 artofproblemsolving.com/community/c3046h1047131__7 – individ Jun 15 '15 at 16:00
• @individ - care to expound your comment to an answer? – nbubis Jun 16 '15 at 6:51
• Write what type of should have functions $x,y$ ? – individ Jun 16 '15 at 7:05
• What do you mean type? They should be real and of the form described – nbubis Jun 16 '15 at 7:06
• This system is the solution. Depending on what type has the left side - you can write some your decision. – individ Jun 16 '15 at 7:14