The article Wikipedia: multiple tangents at a point suggests that in order to find all the tangents at an implicit curve:
$ (x^2+(y-1)^2-1)(x-y) = 0 $
Which has a branch point on (0,0). To find its tangents We must exclude the lowest polynomial degree (2):
$ -2y(x-y) = 0$
The solutions to these include the direction tangents to the point (IE, the vectors (c,0) and (c,c) where c is a constant).
I would like to know if the same approach can be applied in the case of intersection curve to multiple surfaces. For example:
$ (x+y^2+z)(x^2+y^2+z) = 0 $
$ x^3+y^2+z(y-z) = 0 $
Create an intersection curve with branch point occuring at (0,0,0). Can I find the tangents to this point by simply taking the lowest polynomial degree of every equation and solving for:
$ z(x+z) = 0 $
$ y^2+z(y-z) = 0 $
with let's say x = 1?