Related papers: A DG $\Gamma$-structure on the Generalized Taylor …
We call a graded connected algebra $R$ effectively coherent, if for every linear equation over $R$ with homogeneous coefficients of degrees at most $d$, the degrees of generators of its module of solutions are bounded by some function…
We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories…
The so called generalized down-up algebras are revisited from a viewpoint of Gr\"obner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $\lambda\omega\ne 0$), and…
We study the classification of $\mathbb{Z}$-DGAs with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert >0$, motivated by computations in algebraic $K$-theory. This classification problem was left open in work of Dwyer, Greenlees,…
We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials.
Let $k$ be an algebraically closed field of characteristic $p>0$, let G=GL_n be the general linear group over $k$, let g=gl_n be its Lie algebra and let $D_s$ be subalgebra of the divided power algebra of g^* spanned by the divided power…
In this work, we study the classification of differential graded algebras over $\mathbb{Z}$ (DGAs) whose homology is $\mathbb{F}_p[x]$, i.e. the polynomial algebra over $\mathbb{F}_p$ on a single generator. This classification problem was…
Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…
We provide fundamental results on positive solutions to parametrized systems of generalized polynomial $\textit{inequalities}$ (with real exponents and positive parameters), including generalized polynomial $\textit{equations}$. In doing…
A restricted $d$th power of an ideal $I$ is obtained by restricting the exponent vectors allowed to appear on the "natural" generating set of $I^d$, for some integer $d$. In this paper, we study homological properties of restricted powers…
In 1999, Grunbaum, Haine and Horozov defined a large family of commutative algebras of ordinary differential operators which have orthogonal polynomials as eigenfunctions. These polynomials are mutually orthogonal with respect to a…
Inspired by Brown's collapsing method (or discrete Morse theory) to obtain a free resolution of $\bbZ$ over the monoid ring $\bbZ M$, we apply algebraic discrete Morse theory to compute the homology groups of Lawvere theories, which is…
We use the lcm-lattice of a monomial ideal to study its minimal free resolutions. A new concept called a Taylor basis of a minimal free resolution is introduced and then used throughout the paper. We give a method of constructing minimal…
To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of…
Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial…
Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual…
The underlying algebra for a noncommutative geometry is taken to be a matrix algebra, and the set of derivatives the adjoint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of…
The aim of the paper is to study relations between polynomial solutions of generalized Moisil-Theodoresco (GMT) systems and polynomial solutions of Hodge-de Rham systems and, using these relations, to describe polynomial solutions of GMT…
We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from…
We introduce gamma structures on regular hypergeometric D--modules in dimension 1 as special one--parametric systems of solutions on the compact subtorus. We note that a balanced gamma product is in the Paley--Wiener class and show that the…