The standard algorithm for extracting square roots can be used to extract the square root of a polynomial of even degree if it has one square root. Also, one can use the method of indeterminate coefficients to extract square roots of polynomials in one variable.
I've tried to use this last process to extract square roots of polynomials in two variables. I've succeed, but I am unable to precise the exact rules behind the method. For example, the following polynomial has, by construction, a square root:
$1089 p^6 z^4 + 726 p^3 w^4 x^2 y z^2 - 66 p^3 x^3 y^6 z^3 + 121 w^8 x^4 y^2 - 22 w^4 x^5 y^7 z + x^6 y^{12} z^2 $
What type of algorithm that works for polynomials in any number of variables should I apply to extract square roots? How exactly such algorithm can be applied to the polynomial written above?
start at highest (total) degree, 20: $$ \left(x^3 y^6 z \right)^2 = x^6 y^{12} z^2 $$
There is no other degree 20 term, and the next lower is degree 17, $-22 w^4 x^5 y^7 z.$ This degree 17 term is a multiple of $x^3 y^6 z$ which means we can put a second term to take care of 17:
$$ \left(x^3 y^6 z -11 w^4 x^2 y \right)^2 = x^6 y^{12} z^2-22 w^4 x^5 y^7 z + 121 w^8 x^4 y^2 $$ meaning that the three right hand terms of the original expression are a square all by themselves, the term we got for free of degree 14.
We are left with highest degree 15 at $-66p^3 x^3 y^6 z^3.$ This is, once again, divisible by $x^3 y^6 z,$ and suggests adding in a term to take care of this 15 and seeing what happens.
$$ \left(x^3 y^6 z -11 w^4 x^2 y -33 p^3 z^2\right)^2 = x^6 y^{12} z^2 - 22 w^4 x^5 y^7 z + 121 w^8 x^4 y^2 - 66 p^3 x^3 y^6 z^3 + 726 p^3 w^4 x^2 y z^2 + 1089 p^6 z^4$$