Point out the main difference or relation between Newton's polygon and Hensel's lemma when it comes to find solution of the two variable polynomial $ f(x,y)=y^6-5xy^5+x^3y^4-7x^2y^2+6x^3+x^4=0$.
$ \text{Using Newton's polygon}:$
Plot the points (exponents $(a,b)$) as given below:
$(6,0), (5,1), (4,3), (2,2), (0,3), (0,4)$.
The plot is given below:
Thus the vertices of the Newton polygon are $ \ (0,3), \ ( 2,2), \ (6,0)$.
The line segment joining these points is $ \ x+2y=6$.
Now the monomial s corresponding to the vertices $ \ (0,3), \ ( 2,2), \ (6,0)$ are
$y^6-7x^2y^2+6x^3=0, .......(1)$.
This can be solved easily.
Divide the equation $(1)$ by $x^3$ both sides,
$ \frac{y^6}{x^3}-7 \frac{y^2}{x}+6=0$
or, $ Y^6-7Y^2+6=0$, where $ \ Y=\frac{y}{\sqrt x}$,
This can be written as
$ (Y^2-1)(Y^2-2)(Y^2+3)=0 \\ \Rightarrow Y=\pm 1, \ \pm \sqrt 2, \ \pm \sqrt{-3} . $
i.e., $ y=\pm \sqrt x, \ y=\pm \sqrt {2x}, \ y=\pm \sqrt{-3x}$.
These are the solutions by using $ \text{Newton Polygon}$.
But what would be the solution by using Hensel's Lemma?
What would be the difference?
Please help me using Hensel's lemma and pointing out the difference or relation in both method.