# Calculating infinitely near points of a curve singularity

Let $$f(x,y)$$ be a polynomial in $$\mathbb{Q}[x,y]$$ with a singular point at the origin $$(0,0)$$. Currently I am writing a Maple program to compute its iterated blow-up embedded resolution as a hierarchy of charts which are linked by coordinate transformations of the type $$[x = x, y = t x]$$ and $$[x = s y, y = y]$$. It works quite straightforward, the greatest difficulty was the bookkeeping and to find a proper order for recursion. See the bottom of the page

http://www.aviduratas.de/compalg.html

for the current version and an example use.

The resolution graphs that I compute from this tree seem to come out right now, but I have problems with computing the (numbers and multiplicites of) infinitely near points of $$(0,0)$$, that is the points of the transform of $$f = 0$$ that lie above $$(0,0)$$ in the collection of charts generated.

For debugging it would be useful to have a computer algebra system that can calculate the (multiplicities of) infinitely near points, or at least to have a list of known correct examples.

At first I thought that for example the Singular system should have such functionality but I could not find it in the manual.

Does anyone know of a software for calculating the infinitely near points that is publicly available on the internet? Maybe someone who wrote for his own use such a routine?

(I came to write this program to solve Hartshorne V, ex. 3.8. While researching the above question I noted the following post:

Configuration of infinitely near points

My own calculations support the view of the OP there that there is a mistake in the exercise, does anyone know more here?)