Abstract
Cylindrical decomposition and false derivatives are used to find the number of distinct real solutions of a polynomial with integral coefficients p(x, x n ), where n is large. The computation time depends only poly normally on the size of the problem, where this is defined to be the maximum of d, log(n) and log(C), where d is the total degree of p(x, y), assumed to be small relative to n, and C is the maximum of the absolute values of the coefficients.
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References
A. Baker, Transcendental Number Theory, Cambridge University Press, 1975
A. Baker, Acta Arithmetica 37 (1980) pp 257–283
R. P. Brent, JACM 23, pp 242–251, 1976
G. E. Collins and R. G. K. Loos, Real Zeros of Polynomials, Computing Supplementum 4 (ed) B. Buchberger, G. E. Collins and R. G. K. Loos) Springer-Verlag, Wien-New York, 1982, pp 83–94
F. Cucker, L. G. Vega, F. Rossello, On Algorithms for real algebraic plane curves, in Effective Methods in Algebraic Geometry, Edited by Teo Mora, Carlo Traverso, Birkhauser, 1991
J. H. Davenport, Computer algebra for cylindrical algebraic decomposition, Bath computer science technical report 88-10
M. Mignotte and M. Waldschmidt, Linear forms in two logarithms, Acta Arith 53, 1989, pp 251–287
M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge University Press, 1989
D. Richardson, Finding roots of equations involving functions defined by first order algebraic differential equations, in Effective Methods in Algebraic Geometry, edited by Teo Mora, Carlo Traverso, Birkhauser, Boston, Basel, Berlin 1991
D. Richardson, Computing (in a bounded part of the plane) the topology of a real curve defined by solutions of algebraic differential equations, submitted to JSC
M. F. Roy, Computation of the topology of a real curve, proceeding of the conference on computational geometry and topology, Sevilla 1987
R. Zipple, Probabalistic algorithms for sparse polynomials, Proceedings EUROSAM 79 (Marseille) Springer LNCS 72 pp 216–226
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© 1993 Birkhäuser Boston
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Richardson, D. (1993). Finding the number of distinct real roots of sparse polynomials of the form p(x,x n). In: Eyssette, F., Galligo, A. (eds) Computational Algebraic Geometry. Progress in Mathematics, vol 109. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2752-6_16
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DOI: https://doi.org/10.1007/978-1-4612-2752-6_16
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