Up till now, I’ve been using power series to parametrize branches:

*x*(*t*) = *a*_{0}+*a*_{1}*t*+*a*_{2}*t*^{2}+ ⋯, *y*(*t*) = *b*_{0}+*b*_{1}*t*+*b*_{2}*t*^{2}+ ⋯

If the branch passes through the origin, then *a*_{0}=*b*_{0}=0. In the last post, we established Facts 4 and 5, *assuming* that *y*(*t*)=*t* for all branches, so *x*(*t*) = *a*_{0}+*a*_{1}*y*+*a*_{2}*y*^{2}+ ⋯.