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?

Look at $0=f(x,y)$ with a magnifying glass $0=f(ε X,ε Y)=ε^3(X^3+ε X^4+Y^3)$ with $X$ and $Y$ always in the range $[-1,1]$. There you see that the higher order term has a coefficient that becomes smaller the closer you look. So in the first order approximation, $Y=-q^kX$, with $q\ne 1$, $q^3=1$ the third root of unity.
Now one try to find the next order of the approximation, $Y=-q^kX+δ$,
$$0=X^3+ε X^4+(-q^kX+δ)^3=ε X^4+3q^{-k}X^2δ+O(δ^2)$$
which gives $δ=-\tfrac13εq^k X^2$, so that now the approximation up to terms in $ε^2$ is
$$Y=-q^k(X+\tfracε3X^2).$$

Of course, in this case with only one term involving $y$, the faster solution is by directly computing the cube root, then
$$y=-q^kx\,\sqrt[3]{1+x}=-q^kx\,\left(1+\tfrac13 x-\tfrac19x^2+\tfrac{5}{81}x^3+\dots+\tbinom{1/3}{k}x^k+\dots\right)$$
