Related papers: Effective computation of centralizers of ODOs

The correspondence between commutative rings of ordinary differential operators and algebraic curves has been extensively and deeply studied since the seminal works of Burchnall-Chaundy in 1923. This paper is an overview of recent…

We consider the classical factorization problem of a third order ordinary differential operator $L-\lambda$, for a spectral parameter $\lambda$. It is assumed that $L$ is an algebro-geometric operator, that it has a nontrivial centralizer,…

Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on…

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations of the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dickey (GD) hierarchies. In this paper, we review the formal construction of the…

The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention…

Let \Y be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study H(\Y), a monoidal DG category that might be regarded as a categorification of the ring of…

The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the…

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal $\mathcal{I}$ of the local ring $\mathcal{O}$ of an $n$-space algebroid curve with several branches. This allows us to determine the semimodule of…

The theory of nilpotent orbits of simple Lie algebras has seen tremendous developments over the past decades. In this context an important role is played by the component group of the stabilizer of a nilpotent element. In this work, the aim…

The aim of the paper is to study the ring of differential operators $\mathcal{D}(A(m))$ on the generalized multi-cusp algebra $A(m)$ where $m\in \mathbb{N}^n$ (of Krull dimension $n$). The algebra $A(m)$ is singular apart from the single…

We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…

Ore operators with polynomial coefficients form a common algebraic abstraction for representing D-finite functions. They form the Ore ring $K(x)[D_x]$, where $K$ is the constant field. Suppose $K$ is the quotient field of some principal…

We investigate the structure of the centralizer and the normalizer of a local analytic or formal differential system at a nondegenerate stationary point, using the theory of Poincar\'e-Dulac normal forms. Our main results are concerned with…

The ring $\text{Diff}_{\mathbf{h}}(n)$ of $\mathbf{h}$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\mathbf{h}$-deformed…

In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible…

We study the space of the solutions $s$ of any system of partial differential equations $ D(j^ks)=0 $ defined by a linear and homogeneous differential operator $ D:J^kE\to F $ of any order $k\geq 1$, which is ``ordinary" (i.e. which is…

We give an algorithm to compute the following cohomology groups on $U = \C^n \setminus V(f)$ for any non-zero polynomial $f \in \Q[x_1, ..., x_n]$; 1. $H^k(U, \C_U)$, $\C_U$ is the constant sheaf on $U$ with stalk $\C$. 2. $H^k(U, \Vsc)$,…

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…

We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…

Block encoding lies at the core of many existing quantum algorithms. Meanwhile, efficient and explicit block encodings of dense operators are commonly acknowledged as a challenging problem. This paper presents a comprehensive study of the…