This paper deals with the representation of the solutions of a polynomial system, and concentrates on the high-dimensional case. Based on the rational univari- ate representation of zero-dimensional polynomial systems...This paper deals with the representation of the solutions of a polynomial system, and concentrates on the high-dimensional case. Based on the rational univari- ate representation of zero-dimensional polynomial systems, we give a new description called rational representation for the solutions of a high-dimensional polynomial sys- tem and propose an algorithm for computing it. By this way all the solutions of any high-dimensional polynomial system can be represented by a set of so-called rational- representation sets.展开更多
Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible c...Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.展开更多
Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮（g） = {（x,y） ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie...Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮（g） = {（x,y） ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250： 473-484], we show that the variety ∮（g） is reducible, and not equidimensional. Irreducible components of ∮（g） and their dimensions are precisely given. As a consequence, the variety ∮（g） is not normal.展开更多
We describe all degenerations of the variety ■3 of Jordan algebras of dimension three over C.In particular,we describe all irreducible components in ■3.For every n we define an n-dimensional rigid“marginal”Jordan ...We describe all degenerations of the variety ■3 of Jordan algebras of dimension three over C.In particular,we describe all irreducible components in ■3.For every n we define an n-dimensional rigid“marginal”Jordan algebra of level one.Moreover,we discuss marginal algebras in associative,alternative,left alternative,non-commutative Jordan,Leibniz and anticommutative cases.展开更多
基金The National Grand Fundamental Research 973 Program (2004CB318000) of China
文摘This paper deals with the representation of the solutions of a polynomial system, and concentrates on the high-dimensional case. Based on the rational univari- ate representation of zero-dimensional polynomial systems, we give a new description called rational representation for the solutions of a high-dimensional polynomial sys- tem and propose an algorithm for computing it. By this way all the solutions of any high-dimensional polynomial system can be represented by a set of so-called rational- representation sets.
基金the National Natural Science Foundation of China (Grant No. 10671009)
文摘Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.
文摘Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮（g） = {（x,y） ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250： 473-484], we show that the variety ∮（g） is reducible, and not equidimensional. Irreducible components of ∮（g） and their dimensions are precisely given. As a consequence, the variety ∮（g） is not normal.
基金supported by FAPESP(16/16445-0,18/15712-0),RFBR(18-31-00001)the President's Program Support of Young Russian Scientists(grant MK-2262.2019.1).
文摘We describe all degenerations of the variety ■3 of Jordan algebras of dimension three over C.In particular,we describe all irreducible components in ■3.For every n we define an n-dimensional rigid“marginal”Jordan algebra of level one.Moreover,we discuss marginal algebras in associative,alternative,left alternative,non-commutative Jordan,Leibniz and anticommutative cases.
