I am reading "Lectures on Étale Cohomology" by J.S. Milne and I am trying to understand example 2.7.
It says that the tangent cone of the affine variety defined by $Y^2=X^3+X^2$ at the origin is defined by $Y^2=X^2$.
I am trying to show this using the following definition. For an affine variety $V= \text{Specm }k[X_1,\ldots X_n]/\mathfrak{a}$ over an algebraically closed field $k$, the tangent cone at the origin P is defined by the ideal $\mathfrak{a}_* =\{f_*:f\in\mathfrak{a}\}$ where, for $f\in k[X_1,\ldots X_n]$, $f_*$ is the homogeneous part of $f$ of lowest degree.
If I understand correctly, Milne seems to be claiming that $\mathfrak{a}_*=(Y^2-X^2)$. However, this seems strange to me as $Y^2-X^2$ is not a factor of the polynomial $Y^2-X^3-X^2$. Maybe I don't understand what he means with "homogeneous part of lowest degree"? Any help is greatly appreciated.
For any polynomial $f\in k[x_1,\cdots,x_n]$ we can write $f=\sum_{i\in\Bbb Z_{\geq 0}} f_i$ where the $f_i$ consists of all the terms of degree $i$. Then the smallest $i$ so that $f_i\neq 0$ is the homogeneous part of lowest degree. In your example, with $f=y^2-x^3-x^2$, we have $f_0=f_1=0$, $f_2=y^2-x^2$, $f_3=-x^3$, and $f_i=0$ for $i>3$, so $f_2=y^2-x^2$ is the homogeneous part of lowest degree.