In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjuga...In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjugate ones. The total number of zeroes does not exceed the degree of the polynomial. We also present a technique for computing the zeroes.展开更多
A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the numbe...A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the number of evaluations of the polynomial)grows no faster than O(max{n^4,n^3log_2(n/ε)}).This work is in response to a question raised in a paper by S.Smale as to the efficiency of piecewise linear methods in solving equations.In comparison with a few results reported,the algorithm under discussion is the only one providing correct multiplicities and the only one employing vector labelling.展开更多
基金sponsored by the National Natural ScienceFunds for Young Scholars (10901166)the Scientific Research Foundation for the Youth Scholars of Sun Yat-SenUniversitythe Research Grant of University of Macao on Applications of Hyper-Complex Analysis (cativo:7560)
文摘In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjugate ones. The total number of zeroes does not exceed the degree of the polynomial. We also present a technique for computing the zeroes.
基金This work is supported in part by the Foundation of Zhongshan University Advanced Research Centrein part by the National Natural Science Foundation of China
文摘A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the number of evaluations of the polynomial)grows no faster than O(max{n^4,n^3log_2(n/ε)}).This work is in response to a question raised in a paper by S.Smale as to the efficiency of piecewise linear methods in solving equations.In comparison with a few results reported,the algorithm under discussion is the only one providing correct multiplicities and the only one employing vector labelling.
