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Mathematics of Computation

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Factoring multivariate polynomials via partial differential equations


Author: Shuhong Gao
Journal: Math. Comp. 72 (2003), 801-822
MSC (2000): Primary 12Y05, 68W30; Secondary 11Y16, 12D05, 13P05
DOI: https://doi.org/10.1090/S0025-5718-02-01428-X
Published electronically: February 28, 2002
MathSciNet review: 1954969
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Abstract: A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. As in Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored, and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.


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Additional Information

Shuhong Gao
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
Email: sgao@math.clemson.edu

DOI: https://doi.org/10.1090/S0025-5718-02-01428-X
Keywords: Polynomial factorization, absolute irreducibility, partial differential equations, Hilbert irreducibility theorem
Received by editor(s): July 19, 2000
Received by editor(s) in revised form: May 8, 2001
Published electronically: February 28, 2002
Additional Notes: The author was supported in part by NSF Grant DMS9970637, NSA Grant MDA904-00-1-0048 and ONR Grant N00014-00-1-0565. Part of the work was done while the author was a member at the Mathematical Sciences Research Institute in Berkeley, CA, USA
Article copyright: © Copyright 2002 American Mathematical Society