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Analyzing the Dual Space of the Saturated Ideal of a Regular Set and the Local Multiplicities of its Zeros
  • Xialiang Li,
  • Wei Niu
Xialiang Li
Dongguan University of Technology - City College

Corresponding Author:[email protected]

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Wei Niu
Beihang University
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In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition of multivariate polynomials, and then propose an algorithm for decomposing a regular set into a finite number of simple sets. From the output of this algorithm, the multiplicities of zeros could be directly read out, and the real solution isolation with multiplicity can also be easily produced. As a main theoretical result of this paper, we analyze the structure of dual space of the saturated ideal generated by a simple set as well as a regular set. Experiments with a preliminary implementation show the efficiency of our method.
12 Mar 2021Submitted to Mathematical Methods in the Applied Sciences
12 Mar 2021Submission Checks Completed
12 Mar 2021Assigned to Editor
19 Mar 2021Reviewer(s) Assigned
03 Sep 2021Review(s) Completed, Editorial Evaluation Pending
05 Sep 2021Editorial Decision: Revise Major
25 Sep 20211st Revision Received
25 Sep 2021Assigned to Editor
25 Sep 2021Submission Checks Completed
28 Sep 2021Reviewer(s) Assigned
29 Sep 2021Review(s) Completed, Editorial Evaluation Pending
04 Jan 2022Editorial Decision: Revise Major
22 Jan 20222nd Revision Received
25 Jan 2022Submission Checks Completed
25 Jan 2022Assigned to Editor
26 Jan 2022Reviewer(s) Assigned
29 Jan 2022Review(s) Completed, Editorial Evaluation Pending
31 Jan 2022Editorial Decision: Accept
Aug 2022Published in Mathematical Methods in the Applied Sciences volume 45 issue 12 on pages 7243-7254. 10.1002/mma.8158