13 research outputs found
Hierarchy of general invariants for bivariate LPDOs
We study invariants under gauge transformations of linear partial
differential operators on two variables. Using results of BK-factorization, we
construct hierarchy of general invariants for operators of an arbitrary order.
Properties of general invariants are studied and some examples are presented.
We also show that classical Laplace invariants correspond to some particular
cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200
Multiple Factorizations of Bivariate Linear Partial Differential Operators
We study the case when a bivariate Linear Partial Differential Operator
(LPDO) of orders three or four has several different factorizations.
We prove that a third-order bivariate LPDO has a first-order left and right
factors such that their symbols are co-prime if and only if the operator has a
factorization into three factors, the left one of which is exactly the initial
left factor and the right one is exactly the initial right factor. We show that
the condition that the symbols of the initial left and right factors are
co-prime is essential, and that the analogous statement "as it is" is not true
for LPDOs of order four.
Then we consider completely reducible LPDOs, which are defined as an
intersection of principal ideals. Such operators may also be required to have
several different factorizations. Considering all possible cases, we ruled out
some of them from the consideration due to the first result of the paper. The
explicit formulae for the sufficient conditions for the complete reducibility
of an LPDO were found also
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
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