32,995 research outputs found
Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra
We present algorithms to factorize weighted homogeneous elements in the first
polynomial Weyl algebra and -Weyl algebra, which are both viewed as a
-graded rings. We show, that factorization of homogeneous
polynomials can be almost completely reduced to commutative univariate
factorization over the same base field with some additional uncomplicated
combinatorial steps. This allows to deduce the complexity of our algorithms in
detail. Furthermore, we will show for homogeneous polynomials that
irreducibility in the polynomial first Weyl algebra also implies irreducibility
in the rational one, which is of interest for practical reasons. We report on
our implementation in the computer algebra system \textsc{Singular}. It
outperforms for homogeneous polynomials currently available implementations
dealing with factorization in the first Weyl algebra both in speed and elegancy
of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table
Macaulay inverse systems revisited
Since its original publication in 1916 under the title "The Algebraic Theory
of Modular Systems", the book by F. S. Macaulay has attracted a lot of
scientists with a view towards pure mathematics (D. Eisenbud,...) or
applications to control theory (U. Oberst,...).However, a carefull examination
of the quotations clearly shows that people have hardly been looking at the
last chapter dealing with the so-called "inverse systems", unless in very
particular situations. The purpose of this paper is to provide for the first
time the full explanation of this chapter within the framework of the formal
theory of systems of partial differential equations (Spencer operator on
sections, involution,...) and its algebraic counterpart now called "algebraic
analysis" (commutative and homological algebra, differential modules,...). Many
explicit examples are fully treated and hints are given towards the way to work
out computer algebra packages.Comment: From a lecture at the International Conference : Application of
Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI
Noncommutative Computer Algebra in the Control of Singularly Perturbed Dynamical Systems
Most algebraic calculations which one sees in linear systems theory, for example in IEEE TAC, involve block matrices and so are highly noncommutative. Thus conventional commutative computer algebra packages, as in Mathematica and Maple, do not address them. Here we investigate the usefulness of noncommutative computer algebra in a particular area of control theory-singularly perturbed dynamic systems-where working with the noncommutative polynomials involved is especially tedious. Our conclusion is that they have considerable potential for helping practitioners with such computations. For example, the methods introduced here take the most standard textbook singular perturbation calculation, [KK086], one step further than had been done previously. Commutative Groebner basis algorithms are powerful and make up the engines in symbolic algebra packages’ Solve commands. Noncommutative Groebner basis algorithms are more recent, but we shall see that they are useful in manipulating the messy sets of noncommutative polynomial equations which arise in singular perturbation calculations. We use the noncommutative algebra package NCAlgebra and the noncommutative Groebner basis package NCGB which runs under it
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