Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAnalogue of Newton-Puiseux series for non-holonomic D-modules and factoring
103
0
0.0
(
0
)
Authors
D.Grigoriev
Ask ChatGPT about the research
No Arabic abstract
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.
rate research
Read More
We study {it non-holonomic} overideals of a left differential ideal $Jsubset F[partial_x, partial_y]$ in two variables where $F$ is a differentially closed field of characteristic zero. The main result states that a principal ideal $J=< P>$ generated by an operator $P$ with a separable {it symbol} $symb(P)$, which is a homogeneous polynomial in two variables, has a finite number of maximal non-holonomic overideals. This statement is extended to non-holonomic ideals $J$ with a separable symbol. As an application we show that in case of a second-order operator $P$ the ideal $<P>$ has an infinite number of maximal non-holonomic overideals iff $P$ is essentially ordinary. In case of a third-order operator $P$ we give few sufficient conditions on $<P>$ to have a finite number of maximal non-holonomic overideals.
On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $mathcal{M}mapstomathcal{M}_{mathrm{reg}}$, called regularization. Recall that $mathcal{M}_{mathrm{reg}}$ is reconstructed from the de Rham complex of $mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.
Let f be a quasi-homogeneous polynomial with an isolated singularity. We compute the length of the D-modules $Df^c/Df^{c+1}$ generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. For 1/f we obtain one more than the reduced genus of the singularity. We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the aforementioned quotient is nonzero when c is a root of the b-function of f (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these D-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.
For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.
In this paper, we show how the non-holonomic control technique can be employed to build completely controlled quantum devices. Examples of such controlled structures are provided.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
