A Fast Algorithm for Computing the Truncated Resultant

Let P and Q be two polynomials in K[x,y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod xk in O~(kd) arithmetic operations in K, where the ~O notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d3) operations before computing its first k coefficients.

[1]  Erich Kaltofen,et al.  On the complexity of computing determinants , 2001, computational complexity.

[2]  Frédéric Lehobey.  Resolvent computations by resultants without extraneous powers , 1997, ISSAC.

[3]  Michael Sagraloff,et al.  On the complexity of solving a bivariate polynomial system , 2012, ISSAC.

[4]  Erich Kaltofen,et al.  On fast multiplication of polynomials over arbitrary algebras , 1991, Acta Informatica.

[5]  Daniel Reischert.  Asymptotically fast computation of subresultants , 1997, ISSAC.

[6]  Leonard H. Soicher.  The computation of Galois groups , 1981 .

[7]  Annick Valibouze,et al.  Using Galois Ideals for Computing Relative Resolvents , 1998, J. Symb. Comput..

[8]  Paul Walton Purdom,et al.  The Analysis of Algorithms , 1995 .

[9]  Laureano González-Vega,et al.  An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve , 1996, J. Complex..

[10]  Joachim von zur Gathen,et al.  Subresultants revisited , 2003, Theor. Comput. Sci..

[11]  Sartaj Sahni,et al.  Analysis of algorithms , 2000, Random Struct. Algorithms.

[12]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[13]  Marina Weber,et al.  Using Algebraic Geometry , 2016 .

[14]  Marc Giusti,et al.  A Gröbner Free Alternative for Polynomial System Solving , 2001, J. Complex..

[15]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

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

[17]  Joachim von zur Gathen,et al.  Computing Frobenius maps and factoring polynomials , 2005, computational complexity.

[18]  Éric Schost,et al.  Algorithms for the universal decomposition algebra , 2012, ISSAC.

[19]  Fabrice Rouillier,et al.  Rational univariate representations of bivariate systems and applications , 2013, ISSAC '13.

[20]  Michael Sagraloff,et al.  On the complexity of computing with planar algebraic curves , 2014, J. Complex..

[21]  Fabrice Rouillier,et al.  Improved algorithm for computing separating linear forms for bivariate systems , 2014, ISSAC.

[22]  Fabrice Rouillier,et al.  Solving bivariate systems using Rational Univariate Representations , 2016, J. Complex..

[23]  Guillaume Moroz,et al.  Numeric certified algorithm for the topology of resultant and discriminant curves , 2014, ArXiv.

[24]  Fabrice Rouillier,et al.  Separating linear forms for bivariate systems , 2013, ISSAC '13.

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

[26]  Fabrice Rouillier,et al.  Improved algorithms for solving bivariate systems via Rational Univariate Representations , 2015 .

[27]  Michael Sagraloff,et al.  An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks , 2010, ALENEX.

[28]  Éric Schost,et al.  Power series solutions of singular (q)-differential equations , 2012, ISSAC.

[29]  Philippe Flajolet,et al.  Fast computation of special resultants , 2006, J. Symb. Comput..

[30]  Xavier Caruso.  Resultants and subresultants of p-adic polynomials , 2015, ArXiv.