NDLI: On factoring multi-variate polynomials over algebraically closed fields (abstract)

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On factoring multi-variate polynomials over algebraically closed fields (abstract)

Content Provider	ACM Digital Library
Author	Yokoyama, K. Noro, M. Takeshima, T.
Abstract	For a problem how to find an extension field over which we can obtain an absolutely irreducible factor, Kaltofen gave an answer in 1983 and explicitly in 1985 by employing analytic argument for showing his answer, and Chistov and Grigor'ev also gave the same answer in 1983 by algebraic arguments. Here, we give an alternative proof for Kaltofen's answer in algebraic way, independently to Chistov and Grigor'ev, and by the benefit of new way, we also give several extensions of his answer and properties of absolutely irreducible factors. We also discuss usage of our results for actual computation of absolutely irreducible factors. Here we restrict ourselves to bi-variate polynomials with integer (or rational) coefficients. First we state Kaltofen's answer.Kaltofen's theorem. For a bi-variate monic polynomial ƒ(x, y) with integer coefficients, if ƒ(x, a) is square free for an integer a, then we can get an irreducible factor of ƒ(x, y) over the algebraic closure of Q by factoring ƒ(x, y) over an algebraic extension field containing a root of ƒ(x, a).Our main results are the following:We give an alternative proof of Kaltofen's theorem by using the fact that for a bi-variate polynomial ƒ(x, y) irreducible over Q, all its absolutely irreducible factors are conjugate with each other and the set of all roots of each irreducible factor becomes an imprimitive block of the action of the Galois group of ƒ(x, y).We specialize the theorem to several coefficient domains such as fields with non zero characteristic.By employing the argument used for an alternative proof, we obtain several useful properties of absolutely irreducible factors, from which we can present several quick test for absolute irreducibility of ƒ(x, y) and information of the degree partition of ƒ(x, y) relating to the degree partition of ƒ(x, a), where a is an integer.We apply the property, information of the degree partition, obtained in (3) to improve existing algorithms for absolute factoring.Thus, our further study on the problem in algebraic way provides several improvements on existing algorithms for factoring multi-variate polynomials.
File Format	PDF
ISBN	0201548925
DOI	10.1145/96877.96966
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	1990-07-01
Publisher Place	New York
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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