A simplified rational representation for positive-dimensional polynomial systems and SHEPWM equations solving

Shang, Baoxin; Zhang, Shugong; Tan, Chang; Xia, Peng

doi:10.1007/s11424-017-6324-0

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…Tan and Zhang [22,24] generalized the rational univariate representation theory to high-dimensional polynomial systems and proposed the rational representation theory. Along this, Shang, et al [19] proposed a simplified rational representation and Xiao et al [27] presented an improvement of the rational representation by introducing minimal Dickson basis proposed by [7]. Similar to rational representation, Schost [18] proposed parametric geometric resolution and Safey El Din et al [17] used rational parametrizations to represent all irreducible components of real algebraic sets.…”

Section: Introductionmentioning

confidence: 99%

Rational Univariate Representation of Zero-Dimensional Ideals with Parameters

Wang

Wei²,

Xiao

et al. 2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently based on the extended subresultant theorem for parametric cases, the separating element corresponding to each branch is chosen with the further partition of parameter space. Finally, with the help of parametric greatest common divisor theory a finite set of the rational univariate representation of zero-dimensional ideals with parameters can be obtained.

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Section: Introductionmentioning

confidence: 99%

Rational Univariate Representation of Zero-Dimensional Ideals with Parameters

Wang

Wei²,

Xiao

et al. 2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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show abstract

“…e idea is to compute quasi-Gröbner basis instead of Gröbner basis. Shang et al [6] used this method to compute independent variable sets for computing rational representation of positive-dimensional ideals. It is noticeable that all computations are in the rational function eld.…”

Section: Introductionmentioning

confidence: 99%

Computing Independent Variable Sets for Polynomial Ideals

Yang

Tan

2022

Journal of Mathematics

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Computing independent variable sets for polynomial ideals plays an important role in solving high-dimensional polynomial equations. The computation of a Gröbner basis for an ideal, with respect to a block lexicographical order in classic methods, is huge, and then an improved algorithm is given. Based on the quasi-Gröbner basis of the extended ideal, a criterion of assigning independent variables is gained. According to the criteria, a maximal independent variable set for a polynomial ideal can be computed by assigning indeterminates gradually. The key point of the algorithm is to reduce dimensions so that the unit of computation is one variable instead of a set, which turns a multivariate problem into a single-variable problem and turns the computation of rational function field into that of the fundamental number field. Hence, the computation complexity is reduced. The algorithm has been analysed by an example, and the results reveal that the algorithm is correct and effective.

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An Improvement of the Rational Representation for High-Dimensional Systems

Xiao

et al. 2020

J Syst Sci Complex

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A simplified rational representation for positive-dimensional polynomial systems and SHEPWM equations solving

Cited by 5 publications

References 10 publications

Rational Univariate Representation of Zero-Dimensional Ideals with Parameters

Rational Univariate Representation of Zero-Dimensional Ideals with Parameters

Computing Independent Variable Sets for Polynomial Ideals

An Improvement of the Rational Representation for High-Dimensional Systems

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