Computation of Bivariate Characteristic Polynomials of Finitely Generated Modules over Weyl Algebras

In this paper we generalize the classical Groebner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely generated module over a Weyl algebra. We also present corresponding algorithms and examples of computation of such polynomials and show that a bivariate dimension polynomial can contain some invariants that are not carried by the Bernstein dimension polynomial. The obtained results are applied to the isomorphism problem for $D$-modules; they have also potential applications to classification problems of differential algebraic groups

Invariants of GL_n(F_q) in polynomials mod Frobenius powers

Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups, one for invariants mod Frobenius powers of the irrelevant ideal, one for cofixed spaces of polynomials.Comment: 28 pages. v2: Added references, and altered discussion in Section 5.3 on module structure of G-cofixed space over G-fixed subalgebra. v3: final version, to appear in Proc. Roy. Soc. Edinburgh

Computation of the Strength of PDEs of Mathematical Physics and their Difference Approximations

We develop a method for evaluation of A. Einstein's strength of systems of partial differential and difference equations based on the computation of Hilbert-type dimension polynomials of the associated differential and difference field extensions. Also we present algorithms for such computations, which are based on the Gr\"obner basis method adjusted for the modules over rings of differential, difference and inversive difference operators. The developed technique is applied to some fundamental systems of PDEs of mathematical physics such as the diffusion equation, Maxwell equations and equations for an electromagnetic field given by its potential. In each of these cases we determine the strength of the original system of PDEs and the strength of the corresponding systems of partial difference equations obtained by forward and symmetric difference schemes. In particular, we obtain a method for comparing two difference schemes from the point of view of their strength

Noetherianity of some degree two twisted commutative algebras

In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FI-modules and Delta-modules. In this paper, we add another example to the growing list: we show that certain degree two twisted commutative algebras are noetherian. This example appears to have some fundamental differences from previous examples, and is therefore especially interesting. Reflective of this, our proof introduces new methods for establishing noetherianity that are likely to be applicable in other situations. The algebras considered in this paper are closely related to the stable representation theory of classical groups, which is one source of motivation for their study.Comment: 21 pages; v2: small corrections and added Example 1.

A survey of qq-holonomic functions

We give a survey of basic facts of $q$-holonomic functions of one or several variables, following Zeilberger and Sabbah. We provide detailed proofs and examples.Comment: 21 pages, late

Multivariate Difference-Differential Dimension Polynomials

In this paper we generalize the Ritt-Kolchin method of characteristic sets and the classical Gr\"obner basis technique to prove the existence and obtain methods of computation of multivariate difference-differential dimension polynomials associated with a finitely generated difference-differential field extension. We also give an interpretation of such polynomials in the spirit of the A. Einstein's concept of strength of a system of PDEs and determine their invariants, that is, characteristics of a finitely generated difference-differential field extension carried by every its dimension polynomial.Comment: 28 page

Torus equivariant D-modules and hypergeometric systems

We formalize, at the level of D-modules, the notion that A-hypergeometric systems are equivariant versions of the classical hypergeometric equations. For this purpose, we construct a functor on a suitable category of torus equivariant D-modules and show that it preserves key properties, such as holonomicity, regularity, and reducibility of monodromy representation. We also examine its effect on solutions, characteristic varieties, and singular loci. When applied to certain binomial D-modules, our functor produces saturations of the classical hypergeometric differential equations, a fact that sheds new light on the D-module theoretic properties of these classical systems.Comment: 32 pages, The discussion of normalized Horn systems in v1 now appears in arXiv:1806.0335

Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation

We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate Kolchin-type dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. In particular, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. As a consequence, it is shown that the reflexive closure of a prime difference polynomial ideal is the inverse image of this ideal under a power of the basic translation. We also discuss applications of our results to the analysis of systems of algebraic difference-differential equations.Comment: 16 page

Lattice-Supported Splines on Polytopal Complexes

We study the module $C^r(\mathcal{P})$ of piecewise polynomial functions of smoothness $r$ on a pure $n$-dimensional polytopal complex $\mathcal{P}\subset\mathbb{R}^n$, via an analysis of certain subcomplexes $\mathcal{P}_W$ obtained from the intersection lattice of the interior codimension one faces of $\mathcal{P}$. We obtain two main results: first, we show that in sufficiently high degree, the vector space $C^r_k(\mathcal{P})$ of splines of degree $\leq k$ has a basis consisting of splines supported on the $\mathcal{P}_W$ for $k\gg0$. We call such splines lattice-supported. This shows that an analog of the notion of a star-supported basis for $C^r_k(\Delta)$ studied by Alfeld-Schumaker in the simplicial case holds. Second, we provide a pair of conjectures, one involving lattice-supported splines, bounding how large $k$ must be so that \mbox{dim}_\mathbb{R} C^r_k(\mathcal{P}) agrees with the formula given by McDonald-Schenck. A family of examples shows that the latter conjecture is tight. The proposed bounds generalize known and conjectured bounds in the simplicial case.Comment: 22 pages, 11 figures. v2 (updated from published version): More examples added, as well as new results for the graded case. Index of summation in Definition 4.1 changed, theorems and proofs updated to reflect thi

The SL_3 Jones polynomial of the trefoil: a case study of qq-holonomic sequences

The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.Comment: 10 pages, 3 figures, 2 Mathematica notebook