Mathematics – Commutative Algebra
Scientific paper
2008-06-03
Journal of Algebra, Vol. 320, No. 2, pp. 633-659, 2008
Mathematics
Commutative Algebra
30 pages. Preliminary versions of this paper have been presented at CASC 2003 (arXiv:0806.0478 [math.AC]) and CASC 2005 (arXiv
Scientific paper
10.1016/j.jalgebra.2007.12.023
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.
Terui Akira
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