I am studying this article, but I did not understood the example of the blow up considering the definition of blow up given.
In this part of the article, there is a definition:
Consider a real analytic manifold $M$ and a closed real analytic submanifold $C$ of codimension $m \geq 1$. Without loss of generality, we may suppose that $C$ is affine and is given by the zero locus of a strong regular sequence $y_1, \dots, y_m$ of real analytic functions on $M$: $C = V(y_1, \dots, y_m)$. Consider the real projective space $P^{m−1}(\mathbb{R})$ and denote by $\theta_1, \dots, \theta_m$ its homogeneous coordinates. Define the blowup $\operatorname{Bl}_C(M) \subset M \times P^{m−1}(\mathbb{R})$ of $M$ along $C$ as the variety of zeroes of the ideal generated by the relations $y_i \theta_j − y_j \theta_i$ for every $i, j = 1, \dots, m$. Define the canonical projection $\pi$ as the projection on the first factor.
And this is the example of the blow up given for the principalization:
"The principalization may take several steps after the desingularization. As an example, consider the polynomial $f$ in $\mathbb{R}[x, y, z]$ defined by $f(x, y, z) = x^6 + y^2$. Its zero locus is the $z$-axis which is a smooth submanifold. In particular, the embedded desingularization is trivial and is realized in the zeroth step. However, the principalization is achieved after three steps: the first step is the blowup $y = y_1 x$, $f_1(x, y_1, z) = x^2(x^4 + y_1^2)$, the second step is the blowup $y_1 = y_2 x$, $f_2(x, y_2, z) = x^4(x^2 + y_2^2)$ and the third step is the blowup $y_2 = y_3 x$, $f_3(x, y_3, z) = x^6(1 + y_3^2)$.
I would like to understand how the definition (more specifically the homogeneous coordinates relations) of the blow up was used to obtain for example the first blow up given.