Algorithms for polynomial GCD computation over algebraic function fields

Let L be an algebraic function field in k ≥ 0 parameters t;1;, ..., t;k;. Let f;1;, f;2; be non-zero polynomials in L[x]. We give two algorithms for computing their gcd. The first, a modular GCD algorithm, is an extension of the modular GCD algorithm of Brown for Z[x;1;,...,x;n;] and Encarnacion for Q(α)[x] to function fields. It is uses rational number and rational function reconstruction and trial division. The second, a fraction-free algorithm, is a modification of the Moreno Maza and Rioboo algorithm for computing gcds over triangular sets. The modification reduces coefficient growth in L to be linear. We show how to extend the modular GCD algorithm to work when the minimal polynomial for L is not irreducible. We give an empirical comparison of the two algorithms using implementations in Maple.

[1]  Mark J. Encarnación.  Computing GCDs of Polynomials over Algebraic Number Fields , 1995, J. Symb. Comput..

[2]  E. Hecke.  Lectures on the Theory of Algebraic Numbers , 1981 .

[3]  Marc Moreno Maza,et al.  Polynomial Gcd Computations over Towers of Algebraic Extensions , 1995, AAECC.

[4]  Michael Kalkbrener,et al.  A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties , 1993, J. Symb. Comput..

[5]  Michael B. Monagan,et al.  On the design and implementation of Brown's algorithm over the integers and number fields , 2000, ISSAC.

[6]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[7]  P. L. Montgomery,et al.  An FFT extension of the elliptic curve method of factorization , 1992 .

[8]  Russell J. Bradford,et al.  Some results on the defect , 1989, ISSAC '89.

[9]  Victor Y. Pan,et al.  Acceleration of Euclidean algorithm and extensions , 2002, ISSAC '02.

[10]  Scott McCallum,et al.  The Computation of Polynomial Greatest Common Divisors Over an Algebraic Number Field , 1989, J. Symb. Comput..

[11]  Michael B. Monagan.  In-place Arithmetic for Polinominals over Zn , 1992, DISCO.

[12]  Mark van Hoeij,et al.  A modular GCD algorithm over number fields presented with multiple extensions , 2002, ISSAC '02.

[13]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[14]  James H. Davenport,et al.  The Bath algebraic number package , 1986, SYMSAC '86.

[15]  Daniel Lazard,et al.  Solving Zero-Dimensional Algebraic Systems , 1992, J. Symb. Comput..

[16]  Paul S. Wang,et al.  A p-adic algorithm for univariate partial fractions , 1981, SYMSAC '81.

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

[18]  James H. Davenport,et al.  P-adic reconstruction of rational numbers , 1982, SIGS.

[19]  Barry M. Trager,et al.  Algebraic factoring and rational function integration , 1976, SYMSAC '76.

[20]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[21]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[22]  Lars Langemyr.  An Asymptotically Fast Probabilitstic Algorithm for Computing Polynominal GCD's over an Algebraic Number Field , 1990, AAECC.

[23]  George E. Collins,et al.  Efficient Rational Number Reconstruction , 1995, J. Symb. Comput..