Sparse Hensel Lifting

A new algorithm is introduced which computes the multivariate leading coefficients of polynomial factors from their univariate images. This algorithm is incorporated into a sparse Hensel lifting scheme and only requires the factorization of a single univariate image. The algorithm also provides the content of the input polynomial in the main variable as a by-product. We show how we can take advantage of this property when computing the GCD of multivariate polynomials by sparse Hensel lifting.

[1]  Erich Kaltofen.  Effective Hilbert Irreducibility , 1985, Inf. Control..

[2]  Richard Zippel.  Newton's iteration and the sparse Hensel algorithm (Extended Abstract) , 1981, SYMSAC '81.

[3]  D. McIlroy.  Algebraic Simplification , 1966, CACM.

[4]  Erich Kaltofen,et al.  Factoring Sparse Multivariate Polynomials , 1983, J. Comput. Syst. Sci..

[5]  Erich Kaltofen.  Computing with polynomials given by straight-line programs I: greatest common divisors , 1985, STOC '85.

[6]  H. I. Epstein.  Using basis computation to determine pseudo-multiplicative independence , 1976, SYMSAC '76.

[7]  Erich Kaltofen,et al.  Computing with polynomials given by straight-line programs II sparse factorization , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[8]  W. S. Brown.  On Euclid's algorithm and the computation of polynomial greatest common divisors , 1971, SYMSAC '71.

[9]  Guy Viry.  Factorisation des Polynomes a Plusieurs Variables , 1980, RAIRO Theor. Informatics Appl..

[10]  David R. Musser,et al.  Multivariate Polynomial Factorization , 1975, JACM.

[11]  Keith O. Geddes,et al.  A Comparison of Algorithms for the Symbolic Computation of Padé Approximants , 1984, EUROSAM.

[12]  Joachim von zur Gathen,et al.  Irreducibility of Multivariate Polynomials , 1985, J. Comput. Syst. Sci..

[13]  Paul S. Wang,et al.  The EEZ-GCD algorithm , 1980, SIGS.

[14]  Paul S. Wang,et al.  Early detection of true factors in univariate polynominal factorization , 1983, EUROCAL.

[15]  Paul S. Wang.  An improved multivariate polynomial factoring algorithm , 1978 .

[16]  David Y. Y. Yun,et al.  The EZ GCD algorithm , 1973, ACM Annual Conference.

[17]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[18]  Erich Kalto.  Computing with Polynomials Given by Straight-Line Programs II Sparse Factorization. , 1985 .

[19]  E. Bareiss.  Sylvester’s identity and multistep integer-preserving Gaussian elimination , 1968 .

[20]  David Y. Y. Yun,et al.  The Hensel Lemma in Algebraic Manipulation , 1973, Outstanding Dissertations in the Computer Sciences.