Parallel sparse multivariate polynomial division

We present a scalable algorithm for dividing two sparse multivariate polynomials represented in a distributed format on shared memory multicore computers. The scalability on the large number of cores is ensured by the lack of synchronizations during the main parallel step. The merge and sorting operations are based on binary heap or tree data structures.

[1]  Stephen C. Johnson,et al.  Sparse polynomial arithmetic , 1974, SIGS.

[2]  Joris van der Hoeven,et al.  On the complexity of multivariate blockwise polynomial multiplication , 2012, ISSAC.

[3]  Andrew Arnold,et al.  Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication , 2015, ISSAC.

[4]  Paul S. Wang,et al.  Parallel Polynomial Operations on SMPs: an Overview , 1996, J. Symb. Comput..

[5]  Richard J. Fateman.  Polynomial Multiplication, Powers and Asymptotic Analysis: Some Comments , 1974, SIAM J. Comput..

[6]  D. M. Hutton,et al.  The Art of Multiprocessor Programming , 2008 .

[7]  Michael B. Monagan,et al.  Sparse polynomial division using a heap , 2011, J. Symb. Comput..

[8]  Michael B. Monagan,et al.  Parallel sparse polynomial division using heaps , 2010, PASCO.

[9]  Michael B. Monagan,et al.  Parallel sparse polynomial multiplication using heaps , 2009, ISSAC '09.

[10]  Carl Ponder.  Parallel Multiplication and Powering of Polynomials , 1991, J. Symb. Comput..

[11]  Jacques Laskar,et al.  Highly Scalable Multiplication for Distributed Sparse Multivariate Polynomials on Many-Core Systems , 2013, CASC.

[12]  Bart De Moor,et al.  The Geometry of Multivariate Polynomial Division and Elimination , 2013, SIAM J. Matrix Anal. Appl..

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

[14]  Sven Köhler,et al.  On the Stability of Fast Polynomial Arithmetic , 2008 .

[15]  Daniel S. Roche.  Chunky and equal-spaced polynomial multiplication , 2010, J. Symb. Comput..

[16]  Ellis Horowitz.  A Sorting Algorithm for Polynomial Multiplication , 1975, JACM.

[17]  Richard J. Fateman.  Comparing the speed of programs for sparse polynomial multiplication , 2003, SIGS.

[18]  Joris van der Hoeven,et al.  Sparse Polynomial Interpolation in Practice , 2015, ACCA.

[19]  Michael B. Monagan,et al.  Polynomial Division Using Dynamic Arrays, Heaps, and Packed Exponent Vectors , 2007, CASC.