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
5.	----
6.	0.03	163.34	36659.08
7.	0.03	0.60	83043.46