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Consider the system of non linear equations $$\begin{cases}x^2y^3+x^3y^2+x^5y+1=a \\ xy^2-2x^2y^4+3x^3y=b\end{cases}$$

How can I prove that for a $(a,b)$ close to $(4,2)$ there is a unique solution $x=f(a,b)$, $y=g(a,b)$ close to $(1,1)$?

Note that when $a=4,b=2$ the system has the solution $x=1,y=1$

codingnight
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    Inverse function theorem (https://en.wikipedia.org/wiki/Inverse_function_theorem). – Jeff Dec 15 '18 at 23:22
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    So if I calculate the Jacobian of the vector-valued function of $\mathbb{R^2}$ defined by $F(x,y)=\begin{pmatrix} x^2y^3+x^3y^2+x^5y+1-a \ xy^2-2x^2y^4+3x^3y-b \end{pmatrix}$ And this Jacobian is non-zero in $x=1,y=1$ that would mean that there is a unique solution $x=f(a,b)$, $y=g(a,b)$ close to $(1,1)$? – codingnight Dec 15 '18 at 23:38
  • Yes, not only that, but the mappings $f$ and $g$ are infinitely differentiable! – Jeff Dec 17 '18 at 16:56

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