Related papers: Proof mining and effective bounds in differential …
Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…
A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterisation of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems…
We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a…
Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of set-valued mappings between function spaces. This paper deals with the computational properties of certain…
We consider the following question: When are rings of differential operators coherent? If $A$ is a finitely generated smooth domain over a field $k$ of characteristic $0$, then the ring $D$ of differential operators on $A$ is a Noetherian…
We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…
In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear…
The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…
Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…
We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via work by…
We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the…
We introduce the notion of numerical functors to generalise Eilenberg & MacLane's polynomial functors to modules over a binomial base ring. After shewing how these functors are encoded by modules over a certain ring, we record a precise…
We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the…
We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological…
We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an…
Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field…
Border bases are a generalization of Gr\"obner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra.…