Bit-size reduction of triangular sets in two and three variables

14 pages•Published: March 27, 2016

Abstract

At ISSAC 2004, Schost and D.
introduced a transformation of triangular lexicographic Groebner bases
generating a radical ideal of dimension zero,
which reduces significantly the bit-size of coefficients.
The case where the triangular lexicographic Groebner basis does not generate a radical ideal
is far more complicated. This work treats the case of n=2 variables, and
in some extent the case of n=3 variables.
It resorts to an extra operation, the squarefree factorization;
nevertheless this operation has low complexity cost.
But as soon as n>2 variables a lack of simple and efficient gcd-like operation
over non-reduced rings prevents to undertake meaningful algorithmic considerations.
An implementation in Maple for the case n=2 confirms the expected
reduction of the expected size coefficients.

Keyphrases: bit size, lagrange interpolation, lexicographic groebner bases, multiplicity, triangular sets

In: James H. Davenport and Fadoua Ghourabi (editors). SCSS 2016. 7th International Symposium on Symbolic Computation in Software Science, vol 39, pages 169-182.

Links:	https://easychair.org/publications/paper/mWD
	https://doi.org/10.29007/nz5x

BibTeX entry

@inproceedings{SCSS2016:Bit_size_reduction_triangular,
  author    = {Tetsuro Yamashita and Xavier Dahan},
  title     = {Bit-size reduction of triangular sets in two and three variables},
  booktitle = {SCSS 2016. 7th International Symposium on  Symbolic Computation in Software Science},
  editor    = {James H. Davenport and Fadoua Ghourabi},
  series    = {EPiC Series in Computing},
  volume    = {39},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/mWD},
  doi       = {10.29007/nz5x},
  pages     = {169-182},
  year      = {2016}}

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