Computational Algebraic Geometry pp 225-233 | Cite as

# Finding the number of distinct real roots of sparse polynomials of the form *p(x,x*^{n})

Conference paper

## 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.

## Keywords

Real Root Decimal Place Algebraic Number Total Degree Complete Decomposition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- A. Baker, Transcendental Number Theory, Cambridge University Press, 1975Google Scholar
- A. Baker, Acta Arithmetica 37 (1980) pp 257–283MathSciNetGoogle Scholar
- R. P. Brent, JACM 23, pp 242–251, 1976CrossRefzbMATHMathSciNetGoogle Scholar
- 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 83–94Google Scholar
- 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, 1991Google Scholar - J. H. Davenport, Computer algebra for cylindrical algebraic decomposition, Bath computer science technical report 88-10Google Scholar
- M. Mignotte and M. Waldschmidt, Linear forms in two logarithms, Acta Arith 53, 1989, pp 251–287zbMATHMathSciNetGoogle Scholar
- M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge University Press, 1989Google Scholar
- 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 1991Google Scholar
- 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 JSCGoogle Scholar
- M. F. Roy, Computation of the topology of a real curve, proceeding of the conference on computational geometry and topology, Sevilla 1987Google Scholar
- R. Zipple, Probabalistic algorithms for sparse polynomials, Proceedings EUROSAM 79 (Marseille) Springer LNCS 72 pp 216–226Google Scholar

## Copyright information

© Birkhäuser Boston 1993