Factoring multivariate polynomials via partial differential equations

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.

History of finite fields

“…In [Gao 2003], Gao designed the first softly quadratic time probabilistic reduction of the factorization problem from two to one variable whenever the characteristic of the coefficient field is zero or sufficiently large. His algorithm makes use of the first algebraic de Rham cohomology group of F [x, y, 1/f (x, y)], as previously used by Ruppert [Ruppert 1986[Ruppert , 1999 for testing the absolute irreducibility.…”

“…Over the field of complex numbers, Bajaj et al [Bajaj et al 1993] obtained the bound [Mumford 1995, Theorem 4.17] of Bertini's theorem. Gao [Gao 2003] proved the bound B(f, S) ≤ 2d 3 /|S| whenever F has characteristic 0 or larger than 2d 2 . Then Chèze pointed out [Chèze 2004, Chapter 1] that the latter bound can be refined to B(f, S) ≤ d(d 2 − 1)/|S| by using directly [Ruppert 1986, Satz C].…”

Regularization and Matrix Computation in Numerical Polynomial Algebra

“…Many authors studied the problem and proposed different approaches [8,7,10,12,13,27,28,40,45,73,74,75,76]. In 2003, Gao [29] proposed a hybrid algorithm and later adapted it as a numerical algorithm [30,46,48] along with Kaltofen, May, Yang and Zhi. Here we elaborate the strategies involved in numerical irreducible factorization from the perspective of matrix computation.…”

Approximate factorization of multivariate polynomials via differential equations

“…The coefficients of f may be known to only a fixed precision. Our algorithms are based on the exact algorithms in [7]. However, our procedures are designed to take as input polynomials with imprecise floating point coefficients.…”

“…Our bounds such as deg( f ) ≤ deg(f ) limit the degrees in the individual variables, that is deg x (f ) ≤ deg x (f ) and deg y (f ) ≤ deg y (f ) (rectangular polynomials). Our algorithms are based on the exact algorithms in [7]. All our methods are numerical and we execute our procedures with floating point scalars.…”