AeCodimension/comparison.md

Time comparison

Two other algorithms for computing Ae-Codimension are known: one is due to Hiroshi Teramoto and Katsusuke Nabeshima, the other due to Deeba Afzal, Gerhard Pfister and Shamsa Kanwal.

Hiroshi Teramoto did a time comparison between his and Pfister's algorithm, computing the Ae-codimension of the seven following map-germs:

1. (x,y) \mapsto (x,xy+y^5+y^7)
2. (x,y) \mapsto (x,xy+y^5)
3. (x,y) \mapsto (x,xy+y^6+y^8+ay^9) with a\in\mathbb R
4. (x,y) \mapsto (x,xy+y^6+y^9)
5. (x,y,u_1,u_2) \mapsto (x^2-y^2 +xu_1+yu_2,u_1,u_2)
6. (x_1,\ldots,x_5) \mapsto (x_1^3+x_2^3+x_3x_1+x_4x_2+(-x_5-x_5^2)x_1^2+x_5x_2^2,x_2x_1,x_3,x_4,x_5)
7. (x_1,\ldots,x_6)\mapsto (x_1^3+x_2^3+x_3x_1+x_4x_2+x_5x_1^2+x_6x_2^2, x_2x_1, x_3, x_4, x_5)

Problem 1: Our algorithm cannot (yet) deal with example 5., since it is a map-germ in \mathbb C^4 \to \mathbb C^3 (dimension of target is smaller than source).

Problem 2: our algorithm says the Ae-codimension of example 4. is 5, but it should have Ae-codimension 4. In this case, it happens that \frac{\mathcal O_2}{\langle x,xy+y^6+y^9\rangle} has dimension 6 as $\mathbb C$-vector space, while \frac{\mathbb C[x,y]}{\langle x,xy+y^6+y^9\rangle} has dimension 9 as $\mathbb C$-vector space. Note: this difference of dimensions also happens with example 3., but in that case our algorithm returns the correct answer.

With this in mind, here is a table comparing the time it takes for each algorithm to return the Ae-codimension of each of the map-germs (except 5.). Time is expressed in seconds.

Ex.	Ours	Hiroshi	Pfister
1.	0.01	0.11	0.11
2.	0.01	0.10	0.03
3.	0.07	0.36	0.03
4.	0.01	0.13	0.11
6.	0.03	163.34	36659.08
7.	0.03	0.60	83043.46