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