This paper deals with Hilbert's 16th problem and its generalizations. The configurations of all closed branches of an algebraic curve of degree n are discussed. The maximum number of sheets for an algebraic eq...This paper deals with Hilbert's 16th problem and its generalizations. The configurations of all closed branches of an algebraic curve of degree n are discussed. The maximum number of sheets for an algebraic equation of degree n and the maximum number of limit cycles for a planar algebraic autonomous system are achieved. The author also considers different generalizations and some related problems.展开更多
Let D be a domain in Cn. It is known that if D is a simply connected bounded domain in C with spherical real analytic boundary D, then every local biholomorphic map at boundary as above extends to a biholomorphic map ...Let D be a domain in Cn. It is known that if D is a simply connected bounded domain in C with spherical real analytic boundary D, then every local biholomorphic map at boundary as above extends to a biholomorphic map from D onto the unit ball in Cn. As a consequence, a local biholomorphic map between D1 and D2 where D1 and D2 are simply connected domains in Cn with spherical real analytic boundaries can extend to a global biholomorphic map from D1 onto D2. If the boundary is algebraic, the simply connected condition in the above result can be dropped. In this note, we show that the above phenomenon is no longer true if domains are in algebraic varieties with isolated singularities.展开更多
Based on Ritt-Wu well ordering principle and Wu’s constructive theory of decomposing a polynomial set into irreducible ascending sets, we show that Hong’s 'Provingby Examples' method suits all theorems of eq...Based on Ritt-Wu well ordering principle and Wu’s constructive theory of decomposing a polynomial set into irreducible ascending sets, we show that Hong’s 'Provingby Examples' method suits all theorems of equation type. Moreover, the de-展开更多
We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and d...We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set, etc., we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in Cμ spline ring.展开更多
Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this proble...Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this problem: Under what conditions the convergence d(f(xk),f(J)) → 0 implies k →∞ d(xk,J)→O. It is shown that such implication takes place if fx is continuous and f(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: If f is a C1 function and either i) J is a compact set or ii) for any bounded set B, f-1(B)is bounded, then f(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove that j(J) is a finite set if f is a polynomial. Hence f(J) is nowhere dense in the polynomial case.展开更多
文摘This paper deals with Hilbert's 16th problem and its generalizations. The configurations of all closed branches of an algebraic curve of degree n are discussed. The maximum number of sheets for an algebraic equation of degree n and the maximum number of limit cycles for a planar algebraic autonomous system are achieved. The author also considers different generalizations and some related problems.
文摘Let D be a domain in Cn. It is known that if D is a simply connected bounded domain in C with spherical real analytic boundary D, then every local biholomorphic map at boundary as above extends to a biholomorphic map from D onto the unit ball in Cn. As a consequence, a local biholomorphic map between D1 and D2 where D1 and D2 are simply connected domains in Cn with spherical real analytic boundaries can extend to a global biholomorphic map from D1 onto D2. If the boundary is algebraic, the simply connected condition in the above result can be dropped. In this note, we show that the above phenomenon is no longer true if domains are in algebraic varieties with isolated singularities.
文摘Based on Ritt-Wu well ordering principle and Wu’s constructive theory of decomposing a polynomial set into irreducible ascending sets, we show that Hong’s 'Provingby Examples' method suits all theorems of equation type. Moreover, the de-
基金Project supported by The National Natural Science Foundation of China No.69973010,19871010 and10271022.
文摘We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set, etc., we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in Cμ spline ring.
基金Supported by the National Natural Science Foundation of China (G69774008, G59837270, G1998020308)and National Key Project.
文摘Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this problem: Under what conditions the convergence d(f(xk),f(J)) → 0 implies k →∞ d(xk,J)→O. It is shown that such implication takes place if fx is continuous and f(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: If f is a C1 function and either i) J is a compact set or ii) for any bounded set B, f-1(B)is bounded, then f(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove that j(J) is a finite set if f is a polynomial. Hence f(J) is nowhere dense in the polynomial case.
