Details
| Original language | English |
|---|---|
| Pages (from-to) | 550-560 |
| Number of pages | 11 |
| Journal | Journal of Symbolic Computation |
| Volume | 46 |
| Issue number | 5 |
| Publication status | Published - May 2011 |
Abstract
Keywords
- Coefficient ideal, Desingularization, Resolution of singularities, Strict transform, Weak transform
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Computational Mathematics
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In: Journal of Symbolic Computation, Vol. 46, No. 5, 05.2011, p. 550-560.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A Modified Coefficient Ideal for Use with the Strict Transform
AU - Frühbis-Krüger, Anne
PY - 2011/5
Y1 - 2011/5
N2 - Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main differences are the use of different notions of transforms during the resolution process and the different use of exceptional divisors in the descent in ambient dimension. In this article, we focus on the first difference. Only the approach using the weak transform has up until now been successfully used in implementations, because the other one requires an explicit stratification by the Hilbert-Samuel function at each step of the algorithm which is highly impractical due to the high complexity of the computation of such a stratification. In this article, a (hybrid-type) algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert-Samuel function at each step of the desingularization process. This new approach is not intended to always be superior to the previously implemented one, instead it has its strengths precisely at the weak point of the other one and is thus a candidate to be joined with it by an appropriate heuristic.
AB - Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main differences are the use of different notions of transforms during the resolution process and the different use of exceptional divisors in the descent in ambient dimension. In this article, we focus on the first difference. Only the approach using the weak transform has up until now been successfully used in implementations, because the other one requires an explicit stratification by the Hilbert-Samuel function at each step of the algorithm which is highly impractical due to the high complexity of the computation of such a stratification. In this article, a (hybrid-type) algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert-Samuel function at each step of the desingularization process. This new approach is not intended to always be superior to the previously implemented one, instead it has its strengths precisely at the weak point of the other one and is thus a candidate to be joined with it by an appropriate heuristic.
KW - Coefficient ideal
KW - Desingularization
KW - Resolution of singularities
KW - Strict transform
KW - Weak transform
UR - http://www.scopus.com/inward/record.url?scp=79951856775&partnerID=8YFLogxK
UR - https://arxiv.org/abs/0812.1776
U2 - 10.1016/j.jsc.2010.10.005
DO - 10.1016/j.jsc.2010.10.005
M3 - Article
AN - SCOPUS:79951856775
VL - 46
SP - 550
EP - 560
JO - Journal of Symbolic Computation
JF - Journal of Symbolic Computation
SN - 0747-7171
IS - 5
ER -