29 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
Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators
We study tight bounds and fast algorithms for LCLMs of several linear
differential operators with polynomial coefficients. We analyze the arithmetic
complexity of existing algorithms for LCLMs, as well as the size of their
outputs. We propose a new algorithm that recasts the LCLM computation in a
linear algebra problem on a polynomial matrix. This algorithm yields sharp
bounds on the coefficient degrees of the LCLM, improving by one order of
magnitude the best bounds obtained using previous algorithms. The complexity of
the new algorithm is almost optimal, in the sense that it nearly matches the
arithmetic size of the output.Comment: The final version will appear in Proceedings of ISSAC 201
Multivariate Subresultants in Roots
We give rational expressions for the subresultants of n+1 generic polynomials
f_1,..., f_{n+1} in n variables as a function of the coordinates of the common
roots of f_1,..., f_n and their evaluation in f_{n+1}. We present a simple
technique to prove our results, giving new proofs and generalizing the
classical Poisson product formula for the projective resultant, as well as the
expressions of Hong for univariate subresultants in roots.Comment: 22 pages, no figures, elsart style, revised version of the paper
presented in MEGA 2005, accepted for publication in Journal of Algebr
Resultant-based Elimination in Ore Algebra
We consider resultant-based methods for elimination of indeterminates of Ore
polynomial systems in Ore algebra. We start with defining the concept of
resultant for bivariate Ore polynomials then compute it by the Dieudonne
determinant of the polynomial coefficients. Additionally, we apply
noncommutative versions of evaluation and interpolation techniques to the
computation process to improve the efficiency of the method. The implementation
of the algorithms will be performed in Maple to evaluate the performance of the
approaches.Comment: An updated (and shorter) version published in the SYNASC '21
proceedings (IEEE CS) with the title "Resultant-based Elimination for Skew
Polynomials
On polynomial submersions of degree and the real Jacobian conjecture in
The main result of this paper is the following version of the real Jacobian
conjecture: "Let be a polynomial map with nowhere zero
Jacobian determinant. If the degree of is less than or equal to , then
is injective". Assume that two polynomial maps from to are
equivalent when they are the same up to affine changes of coordinates in the
source and in the target. We completely classify the polynomial submersions of
degree with at least one disconnected level set up to this equivalence,
obtaining four classes. Then, analyzing the half-Reeb components of the
foliation induced by a representative of each of these classes, we prove
there is not a polynomial such that the Jacobian determinant of the map
is nowhere zero. Recalling that the real Jacobian conjecture is true
for maps when all the level sets of are connected, we conclude
the proof of the main result
Scalar q-subresultants and Dickson matrices
Following the ideas of Ore and Li we study q-analogues of scalar subresultants and show how these results can be applied to determine the rank of a GF(q)-linear transformation f of GF(q^n). As an application we show how certain minors of the Dickson matrix D(f), associated with f, determine the rank of D(f) and hence the rank of f