34 lines
1.9 KiB
Markdown
34 lines
1.9 KiB
Markdown
# Time comparison
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Two other algorithms for computing Ae-Codimension are known: one is due to Hiroshi Teramoto and Katsusuke Nabeshima, the other due to
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Deeba Afzal, Gerhard Pfister and Shamsa Kanwal.
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Hiroshi Teramoto did a time comparison between his and Pfister's algorithm, computing the Ae-codimension of the seven following map-germs:
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1. $(x,y) \mapsto (x,xy+y^5+y^7)$
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1. $(x,y) \mapsto (x,xy+y^5)$
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1. $(x,y) \mapsto (x,xy+y^6+y^8+ay^9)$ with $a\in\mathbb R$
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1. $(x,y) \mapsto (x,xy+y^6+y^9)$
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1. $(x,y,u_1,u_2) \mapsto (x^2-y^2 +xu_1+yu_2,u_1,u_2)$
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1. $(x_1,\ldots,x_5) \mapsto (x_1^3+x_2^3+x_3x_1+x_4x_2+(-x_5-x_5^2)x_1^2+x_5x_2^2,x_2x_1,x_3,x_4,x_5)$
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1. $(x_1,\ldots,x_6)\mapsto (x_1^3+x_2^3+x_3x_1+x_4x_2+x_5x_1^2+x_6x_2^2, x_2x_1, x_3, x_4, x_5)$
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**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).
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**Problem 2:** our algorithm says the Ae-codimension of example 4. is 5, but it should have Ae-codimension 4.
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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.
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**Note**: this difference of dimensions also happens with example 3., but in that case our algorithm returns the correct answer.
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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.).
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Time is expressed in seconds.
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| Ex. | Ours | Hiroshi | Pfister |
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| ------- | ---- | ------- | ------- |
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| 1. | 0.01 | 0.11 | 0.11 |
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| 2. | 0.01 | 0.10 | 0.03 |
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| 3. | 0.07 | 0.36 | 0.03 |
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| 4. | 0.01 | 0.13 | 0.11 |
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| 5. | ---- | |
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| 6. | 0.03 | 163.34 | 36659.08 |
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| 7. | 0.03 | 0.60 | 83043.46 |
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