Factoring Rational Polynomials over the Complex Numbers

NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean ci...

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Veröffentlicht in:SIAM journal on computing Jg. 22; H. 2; S. 318 - 331
Hauptverfasser: Bajaj, Chanderjit, Canny, John, Garrity, Thomas, Warren, Joe
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.1993
Schlagworte:
Algebra
Algorithms
Computer science
Exact sciences and technology
Field theory and polynomials
Geometry
Mathematics
Polynomials
Sciences and techniques of general use
Variables
Polynomial
Critical value
N variable function
Singular point
Factorization
Algebraic geometry
Complex number
ISSN:0097-5397, 1095-7111
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Zusammenfassung:NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables ($d$, $c$, and $n$, respectively). If $n$ is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor. After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of $(P(z_1 ,z_2 ) = 0)$ correspond to the connected components of the real surface (or complex curve) $P(z_1 ,z_2 ) = 0$ minus its singular points. In finding the number of connected components of the surface $P = 0$, the surface is projected to the the $z_2 $-plane. The singular points of $P(z_1 ,z_2 )$ lie over the projection's critical values. The inverse image of a grid isolating the critical values in the $z_2 $-plane lifts to a one-dimensional real curve skeleton on the surface $(P = 0)$ whose number of connected components is precisely the number of connected components of $P = 0$ minus its singular points. The connectivity of this curve skeleton is constructed symbolically using Sturm sequences associated with the various polynomials defining these maps. Given the number of irreducible factors and their degrees, the actual factors can be reconstructed using the recent result of Neff [Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 152-162] on finding zeros of one-variable polynomials in NC.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0097-5397
1095-7111
DOI:10.1137/0222024