On Zeros of Algebraic Equations — An Application of Ritt Principle

Wu, Wei

doi:10.1142/9789812791085_0013

Cited by 38 publications

(32 citation statements)

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“…Notice that the following ZDT given in [19] is still valid and the proof is also the same as that in [19].…”

Section: Zero Decomposition Theorems In Rmentioning

confidence: 81%

“…According to [1,19], the above procedure can be exhibited in the form of the procedure (8) as below:…”

Section: Well-ordering Principlesmentioning

confidence: 99%

“…Besides the concept of ascending chain, we also use the concept of Wu chain defined in [19] and the concept of weak chain defined in [20] in our program. In order to save space, we use SZDD [21] to represent Boolean polynomials.…”

Section: Introductionmentioning

confidence: 99%

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A characteristic set method for solving boolean equations and applications in cryptanalysis of stream ciphers*

2008

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This paper presents a characteristic set method for solving Boolean equations, which is more efficient and has better properties than the general characteristic set method. In particular, the authors give a disjoint and monic zero decomposition algorithm for the zero set of a Boolean equation system and an explicit formula for the number of solutions of a Boolean equation system. The authors also prove that a characteristic set can be computed with a polynomial number of multiplications of Boolean polynomials in terms of the number of variables. As experiments, the proposed method is used to solve equations from cryptanalysis of a class of stream ciphers based on nonlinear filter generators. Extensive experiments show that the method is quite effective.

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“…Notice that the following ZDT given in [19] is still valid and the proof is also the same as that in [19].…”

Section: Zero Decomposition Theorems In Rmentioning

confidence: 81%

“…According to [1,19], the above procedure can be exhibited in the form of the procedure (8) as below:…”

Section: Well-ordering Principlesmentioning

confidence: 99%

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A characteristic set method for solving boolean equations and applications in cryptanalysis of stream ciphers*

2008

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“…. , p n is zero-dimensional (with a finite number of complex solutions), then it is well known that the Ritt-Wu method, Gröbner basis methods or subresultant method can be used to transform the system of equations into one or more systems in the triangular form (see [15,[17][18][19][20][21][22]). Therefore, a zero-dimensional parametric SAS (that is, the parametric SAS is generally zero-dimensional for almost all the parametrics's values) can be transformed into one or more systems in the following form:…”

Section: Basic Definitionsmentioning

confidence: 99%

“…Use the Ritt-Wu method, Gröbner basis methods or subresultant methods to transform the parametric SASS i into one or more parametric system(s) of the form TSA (see [15,[17][18][19][20][21][22]). Then, for each component, check whether it is a regular parametric TSA and, if not, transform it into regular TSA by WR algorithm [17,18] and the method described in [10,Theorem 5.3] or Wang's algorithm (see [21]).…”

Section: Real Zeros and Its Distributionsmentioning

confidence: 99%

Real zeros of the zero-dimensional parametric piecewise algebraic variety

Lai

Wang

2009

Sci. China Ser. A-Math.

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The piecewise algebraic variety is the set of all common zeros of multivariate splines.We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semialgebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.

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Analyzing the dual space of the saturated ideal of a regular set and the local multiplicities of its zeros

Wei²

2022

Math Methods in App Sciences

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In this paper, a method is proposed to analyze the structure of dual space of the saturated ideal generated by a regular set and the local multiplicities of its zeros. In detail, 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.

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On Zeros of Algebraic Equations — An Application of Ritt Principle

Cited by 38 publications

References 0 publications

A characteristic set method for solving boolean equations and applications in cryptanalysis of stream ciphers*

A characteristic set method for solving boolean equations and applications in cryptanalysis of stream ciphers*

Real zeros of the zero-dimensional parametric piecewise algebraic variety

Analyzing the dual space of the saturated ideal of a regular set and the local multiplicities of its zeros

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