Tangents at singularities Given an implicit polynomial function $f$ with singularity at the origin. How do I find the tangents to the curve at the point?
Wikipedia says that ignore all the terms except the ones with lowest degree. Why is this true?
Take for example the curve $x^3+x^4+y^3=0$. What are the tangents at the origin?

Consider the curve $x^3 + y^3 + x^4 = 0$.  For $(x, y)$ close to zero, the fourth-order term is negligible compared to the third-order terms, so near the origin the graph will strongly resemble that of $x^3 + y^3 = 0$.  This factors as $(x+y)(x^2 - xy + y^2)$, so there will be a tangent line $x = -y$ and a couple tangent lines with complex slope.
To make this a bit more formal, we can use Hensel lifting to actually factor the expression $x^3 + y^3 + x^4$ as
$$(x + y + \cdots)\left(x - \left[\frac{1}{2} + \frac{\sqrt{3}}{2}i\right]y + \cdots\right)\left(x - \left[\frac{1}{2} - \frac{\sqrt{3}}{2}i\right]y + \cdots\right),$$
where each term is a power series.  We can therefore view this algebraic curve as the union of three distinct analytic curves, each of which has its own tangent line at the origin.
That's the idea, anyway.  If you want to do things in a more algebraic setting, it becomes a commutative algebra problem.  But you should understand this way first.
(Also, to get a feel, it probably helps to start by considering a curve whose tangents at the origin have real slope, so that you can actually look at a picture.  Try $x^2 - y^2 - x^4 = 0$ instead, for instance.)
