In [1], we construct singular varieties associated to a polynomial mapping where such that if G is a local submersion but is not a fibration, then the 2-dimensional homology and intersection homology (with total perve...In [1], we construct singular varieties associated to a polynomial mapping where such that if G is a local submersion but is not a fibration, then the 2-dimensional homology and intersection homology (with total perversity) of the variety are not trivial. In [2], the authors prove that if there exists a so-called very good projection with respect to the regular value of a polynomial mapping , then this value is an atypical value of G if and only if the Euler characteristic of the fibers is not constant. This paper provides relations of the results obtained in the articles [1] and [2]. Moreover, we provide some examples to illustrate these relations, using the software Maple to complete the calculations of the examples. We provide some discussions on these relations. This paper is an example for graduate students to apply a software that they study in the graduate program in advanced researches.展开更多
In [1], Guillaume and Anna Valette associate singular varieties V<sub>F </sub>to a polynomial mapping . In the case , if the set K<sub>0</sub>(F) of critical values of F is empty, then F is not...In [1], Guillaume and Anna Valette associate singular varieties V<sub>F </sub>to a polynomial mapping . In the case , if the set K<sub>0</sub>(F) of critical values of F is empty, then F is not proper if and only if the 2-dimensional homology or intersection homology (with any perversity) of VF </sub>is not trivial. In [2], the results of [1] are generalized in the case where n≥3, with an additional condition. In this paper, we prove that for a class of non-proper generic dominant polynomial mappings, the results in [1] and [2] hold also for the case that the set K<sub>0</sub>(F) is not empty.展开更多
We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappingsof degree 2, using the definition of the so-called “fa?ons” in [2]. We obtain a classification theorem for the asymptotic se...We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappingsof degree 2, using the definition of the so-called “fa?ons” in [2]. We obtain a classification theorem for the asymptotic sets of dominant polynomial mappingsof degree 2. This algorithm can be generalized for the dominant polynomial mappingsof degree d, with any (n,d)∈(N*)2.展开更多
We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure...We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure of polynomial maps H to H=(H_(1)(x_(1),x_(2),…,x_(n)),b_(3)x_(3)+…+b_(n)x_(n)+H^((0))_(2)(x_(2)),H_(3)(x_(1),x_(2)),…,H_(n)(x_(1),x_(2))).展开更多
This paper presents solutions to the function generation problems of Stephenson Ⅲ six-link mechanism with the maximum precision positions for the first time using homotopy method with an improved path-tracking scheme...This paper presents solutions to the function generation problems of Stephenson Ⅲ six-link mechanism with the maximum precision positions for the first time using homotopy method with an improved path-tracking scheme. The new path-tracking scheme is based on the characteristics of the zero sets of homotopy function for polynomial mapping and thus the tracking speed is dramatically increased.展开更多
Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mappi...Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mapping （see Ning et al. （2017））. In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan＇s theorem.展开更多
Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = ...Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .展开更多
In this paper, we describe the structure of quadratic homogeneous polynomial maps F = X ＋ H with JH3 = 0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F = X ＋ H is lin...In this paper, we describe the structure of quadratic homogeneous polynomial maps F = X ＋ H with JH3 = 0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F = X ＋ H is linearly triangularizable.展开更多
文摘In [1], we construct singular varieties associated to a polynomial mapping where such that if G is a local submersion but is not a fibration, then the 2-dimensional homology and intersection homology (with total perversity) of the variety are not trivial. In [2], the authors prove that if there exists a so-called very good projection with respect to the regular value of a polynomial mapping , then this value is an atypical value of G if and only if the Euler characteristic of the fibers is not constant. This paper provides relations of the results obtained in the articles [1] and [2]. Moreover, we provide some examples to illustrate these relations, using the software Maple to complete the calculations of the examples. We provide some discussions on these relations. This paper is an example for graduate students to apply a software that they study in the graduate program in advanced researches.
文摘In [1], Guillaume and Anna Valette associate singular varieties V<sub>F </sub>to a polynomial mapping . In the case , if the set K<sub>0</sub>(F) of critical values of F is empty, then F is not proper if and only if the 2-dimensional homology or intersection homology (with any perversity) of VF </sub>is not trivial. In [2], the results of [1] are generalized in the case where n≥3, with an additional condition. In this paper, we prove that for a class of non-proper generic dominant polynomial mappings, the results in [1] and [2] hold also for the case that the set K<sub>0</sub>(F) is not empty.
文摘We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappingsof degree 2, using the definition of the so-called “fa?ons” in [2]. We obtain a classification theorem for the asymptotic sets of dominant polynomial mappingsof degree 2. This algorithm can be generalized for the dominant polynomial mappingsof degree d, with any (n,d)∈(N*)2.
基金Supported by the National Natural Science Foundation of China(Grant No.11601146,11871241)the Natural Science Foundation of Hunan Province(Grant No.2016JJ3085)the Construct Program of the Key Discipline in Hunan Province.
文摘We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure of polynomial maps H to H=(H_(1)(x_(1),x_(2),…,x_(n)),b_(3)x_(3)+…+b_(n)x_(n)+H^((0))_(2)(x_(2)),H_(3)(x_(1),x_(2)),…,H_(n)(x_(1),x_(2))).
基金Supported by the National Natural Science Foundation of China (No. 59975077)by SWJTU Science Foundation (No. 1999XM07).
文摘This paper presents solutions to the function generation problems of Stephenson Ⅲ six-link mechanism with the maximum precision positions for the first time using homotopy method with an improved path-tracking scheme. The new path-tracking scheme is based on the characteristics of the zero sets of homotopy function for polynomial mapping and thus the tracking speed is dramatically increased.
基金supported by National Natural Science Foundation of China(Grant Nos.11501058 and 11431013)the Fundamental Research Funds for the Central Universities(Grant No.0208005202035)Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZDY-SSW-SYS001)
文摘Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mapping （see Ning et al. （2017））. In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan＇s theorem.
基金Supported by National Natural Science Foundation of China(Grant Nos.11801572,11688101)。
文摘Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .
文摘In this paper, we describe the structure of quadratic homogeneous polynomial maps F = X ＋ H with JH3 = 0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F = X ＋ H is linearly triangularizable.
