Fast rational function reconstruction

Let <i>F</i> be a field and let <i>f</i> and <i>g</i> be polynomials in <i>F[t]</i> satisfying deg <i>f</i> > deg <i>g</i>. Recall that on input of <i>f</i> and <i>g</i> the extended Euclidean algorithm computes a sequence of polynomials <i>(s<sub>i</sub>, t<sub>i</sub>, r<sub>i</sub>)</i> satisfying <i>s<sub>i</sub>f + t<sub>i</sub>g = r<sub>i</sub></i>. Thus for <i>i</i> with gcd<i>(t<sub>i</sub>, f)</i> = 1, we obtain rational functions <i>r<sub>i</sub>/t<sub>i</sub></i> ∈ <i>F(t)</i> satisfying <i>r<sub>i</sub>/t<sub>i</sub> ≡ g</i> (mod <i>f</i>).In this paper we modify the fast extended Euclidean algorithm to compute the smallest <i>r<sub>i</sub>/t<sub>i</sub></i>, that is, an <i>r<sub>i</sub>/t<sub>i</sub></i> minimizing deg <i>r<sub>i</sub></i> + deg <i>t<sub>i</sub></i>. This means that in an output sensitive modular algorithm when we are recovering rational functions in <i>F(t)</i> from their images modulo <i>f(t)</i> where <i>f(t)</i> is increasing in degree, we can recover them as soon as the degree of <i>f</i> is large enough and we can do this fast.We have implemented our modified fast Euclidean algorithm for <i>F = Z<sub>p</sub>, p</i> a word sized prime, in Java. Our fast algorithm beats the ordinary Euclidean algorithm around degree 200. This has application to polynomial gcd computation and linear algebra over <i>Z<sub>p</sub>(t)</i>.

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

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

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

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

[5]  Joachim von zur Gathen,et al.  Analysis of Euclidean Algorithms for Polynomials over Finite Fields , 1990, J. Symb. Comput..

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

[7]  T. Takeshima,et al.  A modular method for Gro¨bner-basis construction over Q and solving system of algebraic equations , 1990 .

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

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

[10]  Roman E. Maeder.  Storage Allocation for the Karatsuba Integer Multipliation Algorithm , 1993, DISCO.

[11]  Keith O. Geddes,et al.  Algorithms for computer algebra , 1992 .

[12]  Mark van Hoeij,et al.  Algorithms for polynomial GCD computation over algebraic function fields , 2004, ISSAC '04.

[13]  David Y. Y. Yun,et al.  Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants , 1980, J. Algorithms.

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

[15]  Erich Kaltofen,et al.  Modular rational sparse multivariate polynomial interpolation , 1990, ISSAC '90.

[16]  Robert T. Moenck,et al.  Fast computation of GCDs , 1973, STOC.

[17]  Michael B. Monagan,et al.  Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction , 2004, ISSAC '04.

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

[19]  Arnold Schönhage,et al.  Schnelle Berechnung von Kettenbruchentwicklungen , 1971, Acta Informatica.

[20]  George E. Collins,et al.  Subresultants and Reduced Polynomial Remainder Sequences , 1967, JACM.

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

[22]  Daniel Lichtblau,et al.  Half-GCD and fast rational recovery , 2005, ISSAC.

[23]  George E. Collins,et al.  Improved Techniques for Factoring Univariate Polynomials , 1996, J. Symb. Comput..

[24]  Carlo Traverso,et al.  Gröbner Trace Algorithms , 1988, ISSAC.

[25]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[26]  Mark J. Encarnación.  On a modular algorithm for computing GCDs of polynomials over algebraic number fields , 1994, ISSAC '94.