相关论文: An algorithmic approach to resolutions
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but…
The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets…
Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…
We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a…
A new algorithm is presented for computing a direct solution to a system of consistent linear equations. It produces a minimum norm particular solution, a generalized inverse (of type {124}), and a null space projection operator. In…
A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials…
Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational article is to introduce the notion of a curved operad, and to develop the operadic calculus…
In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…
Given a standard graded algebra over a field, we consider the relationship between G-quadraticity and the existence of a Koszul filtration. We show that having a quadratic Gr\"obner basis implies the existence of a Koszul filtration for…
A type of iterative orthogonally accumulated projection methods for solving linear system of equations are proposed in this paper. This type of methods are applications of accumulated projection(AP) technique proposed recently by authors.…
We discuss certain homological properties of graded algebras whose trivial modules admit non-pure resolutions. Such algebras include both of Artin-Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module…
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
Projection algorithms are well known for their simplicity and flexibility in solving feasibility problems. They are particularly important in practice due to minimal requirements for software implementation and maintenance. In this work, we…
We consider the partition function of beta-gamma systems in curved space of the type discussed by Nekrasov and Witten. We show how the Koszul resolution theorem can be applied to the computation of the partition functions and to characters…