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IPM
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“School of Mathematics”

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Paper   IPM / M / 13481
School of Mathematics
  Title:   Effective computation of radical of ideals and its application to invariant theory
  Author(s):  A. Hashemi
  Status:   Published
  Journal: LNCS
  Vol.:  8592
  Year:  2014
  Pages:   382-389
  Supported by:  IPM
  Abstract:
The most expensive part of the known algorithms in the calculation of primary fundamental invariants (of rings of polynomial invariants of finite linear groups over an arbitrary field) is the computation of the radicals of complete intersection ideals. Thus, in this paper, we develop effective methods for such calculation. For this purpose, we introduce first a new notion of genericity (called D-quasi stable position) and exhibit a novel deterministic algorithm to put an ideal in Noether position (we show that this new notion of genericity is equivalent to N ther position). Then, we use this algorithm and also the algorithm due to Krick and Logar (to compute radicals of ideals) to present an efficient algorithm to calculate the radical of a complete intersection ideal. Furthermore, we apply this algorithm, to improve the classical methods of computing primary invariants which are based on radical computation. Finally, we have implemented in Maple the mentioned algorithms (to put an ideal in Noether position, to compute the radical of ideals and also primary invariants) and compare the proposed algorithms, via a set of benchmarks, with the corresponding functions in Maple and Magma. The experiments we made seem to show that these first implementations are already more efficient than the corresponding functions of Maple and Magma.


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