Related papers: Constructive $D$-module Theory with \textsc{Singul…
A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first order differential calculus on such a…
Fix a manifold M, and let V be an infinite dimensional Lie algebra of vector fields on M. Assume that V contains a finite dimensional semisimple maximal subalgebra A, the projective or conformal subalgebra. A projective or conformal…
Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…
The semina\"ive algorithm can materialise all consequences of arbitrary datalog rules, and it also forms the basis for incremental algorithms that update a materialisation as the input facts change. Certain (combinations of) rules, however,…
We study the invariant algebraic D-modules on an affine variety under the action of an algebraic group.For linear algebraic groups with the multiplication action by themselves, such D-modules correspond to representations of their Lie…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families…
In this paper, we derive a new computational algorithm for Barrett technique for modular polynomial multiplication, termed BA-P. BA-P is then applied to a new residue arithmetic based Barrett algorithm for modular polynomial multiplication…
We begin with modular form periods, a focal point of several Yuri Manin's works. The similarity is discussed between the corresponding zeta-polynomials and superpolynomials of algebraic links, closely related to Khovanov-Rozansky…
We describe the roots of the Bernstein-Sato polynomial of a monomial ideal using reduction mod p and invariants of singularities in positive chracteristic. We give in this setting a positive answer to a problem of Takagi, Watanabe and the…
We introduce and study a derived version $\mathbf L\mathrm{Bin}$ of the binomial monad on the unbounded derived category $\mathscr D(\mathbb Z)$ of $\mathbb Z$-modules. This monad acts naturally on singular cohomology of any topological…
We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta…
We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated…
Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings…
In~\cite{algorithmic} was given an algorithm that computes arithmetical structures on matrices. We use some of the ideas contained there to get an algorithm that computes arithmetical structures over dominated polynomials. A dominated…
The Waring Problem over polynomial rings asks for how to decompose an homogeneous polynomial of degree $d$ as a finite sum of $d^{th}$ powers of linear forms. First, we give a constructive method to obtain a real Waring decomposition of any…
The most singular residues of the standard meromorphic family of trilinear conformally invariant forms on $\mathcal C^\infty_c(\mathbb R^d)$ are computed. Their expression involves covariant bidifferential operators (generalized Rankin…
We give an algorithm to compute the local $b$ function. In this algorithm, we use the Mora division algorithm in the ring of differential operators and an approximate division algorithm in the ring of differential operators with power…
Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some…
We present a new algorithm for computing a $\mu$-basis of the syzygy module of $n$ polynomials in one variable over an arbitrary field $\mathbb{K}$. The algorithm is conceptually different from the previously-developed algorithms by Cox,…