3 research outputs found
Improved Polynomial Remainder Sequences for Ore Polynomials
Polynomial remainder sequences contain the intermediate results of the
Euclidean algorithm when applied to (non-)commutative polynomials. The running
time of the algorithm is dependent on the size of the coefficients of the
remainders. Different ways have been studied to make these as small as
possible. The subresultant sequence of two polynomials is a polynomial
remainder sequence in which the size of the coefficients is optimal in the
generic case, but when taking the input from applications, the coefficients are
often larger than necessary. We generalize two improvements of the subresultant
sequence to Ore polynomials and derive a new bound for the minimal coefficient
size. Our approach also yields a new proof for the results in the commutative
case, providing a new point of view on the origin of the extraneous factors of
the coefficients
Computing greatest common divisor of several parametric univariate polynomials via generalized subresultant polynomials
In this paper, we tackle the following problem: compute the gcd for several
univariate polynomials with parametric coefficients. It amounts to partitioning
the parameter space into ``cells'' so that the gcd has a uniform expression
over each cell and constructing a uniform expression of gcd in each cell. We
tackle the problem as follows. We begin by making a natural and obvious
extension of subresultant polynomials of two polynomials to several
polynomials. Then we develop the following structural theories about them.
1. We generalize Sylvester's theory to several polynomials, in order to
obtain an elegant relationship between generalized subresultant polynomials and
the gcd of several polynomials, yielding an elegant algorithm.
2. We generalize Habicht's theory to several polynomials, in order to obtain
a systematic relationship between generalized subresultant polynomials and
pseudo-remainders, yielding an efficient algorithm.
Using the generalized theories, we present a simple (structurally elegant)
algorithm which is significantly more efficient (both in the output size and
computing time) than algorithms based on previous approaches