From 0ba4cf8dcfd602695dfcf1f73c7f34e91e558411 Mon Sep 17 00:00:00 2001 From: Nelynx Date: Thu, 6 Feb 2025 12:21:12 +0100 Subject: [PATCH] fill table --- comparison.md | 32 ++++++++++++++++++++++++++++++++ 1 file changed, 32 insertions(+) diff --git a/comparison.md b/comparison.md index e69de29..274ebea 100644 --- a/comparison.md +++ b/comparison.md @@ -0,0 +1,32 @@ +# 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. $(x,y) \mapsto (x,xy+y^5+y^7)$ +1. $(x,y) \mapsto (x,xy+y^5)$ +1. $(x,y) \mapsto (x,xy+y^6+y^8+ay^9)$ with $a\in\mathbb R$ +1. $(x,y) \mapsto (x,xy+y^6+y^9)$ +1. $(x,y,u_1,u_2) \mapsto (x^2-y^2 +xu_1+yu_2,u_1,u_2)$ +1. $(x_1,\ldots,x_5) \mapsto (x_1^3+x_2^3+x_3x_1+x_4x_2+(-x_5-x_5^2)x_1^2+x_5x_2^2,x_2x_1,x_3,x_4,x_5)$ +1. $(x_1,\ldots,x_6)\mapsto (x_1^3+x_2^3+x_3x_1+x_4x_2+x_5x_1^2+x_6x_2^2, x_2x_1, x_3, x_4, x_5)$ + +**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 |