Related papers: On implicational bases of closure systems with uni…

In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings. Let $(K, v)$ be a discrete valued field with valuation ring $\mathcal{O}$ and let $\mathfrak{m}$ be the maximal ideal.…

Central issues of the Dirac constraint formalism are discussed in relation to the algorithmic methods of commutative algebra based on the Groebner basis techniques. For a wide class of finite dimensional polynomial degenerate Lagrangian…

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce…

We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…

For $d \geq 2, \ D \geq 1$, let $\mathscr{P}_{d,D}$ denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms. We prove that for any Calder\'{o}n-Zygmund kernel, $K$, the maximally modulated…

This paper addresses problems on the structural design of control systems taking explicitly into consideration the possible application to large-scale systems. We provide an efficient and unified framework to solve the following major…

In this paper we design a new primal-dual algorithm for the classic discrete optimization problem of maximizing a monotone submodular function subject to a cardinality constraint achieving the optimal approximation of $(1-1/e)$. This…

We develop an efficient $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and…

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem min |sigma(F(X)-G)|_alpha + |C(X)- d|_beta subject to A(X)-b in Q; where sigma(X) denotes the vector of singular values of X,…

Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous…

A characteristic pair is a pair (G,C) of polynomial sets in which G is a reduced lexicographic Groebner basis, C is the minimal triangular set contained in G, and C is normal. In this paper, we show that any finite polynomial set P can be…

A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K. The recent algorithms for computing them…

The renormalization group has proven to be a very powerful tool in physics for treating systems with many length scales. Here we show how it can be adapted to provide a new class of algorithms for discrete optimization. The heart of our…

In this paper, we characterized the relationship between Groebner bases and u-bases: any minimal Groebner basis of the syzygy module for n univariate polynomials with respect to the term-over-position monomial order is its u-basis.…

For nonexpansive fixed-point problems, Halpern's method with optimal parameters, its so-called H-dual algorithm, and in fact, an infinite family of algorithms containing them, all exhibit the exactly minimax optimal convergence rates. In…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

Given polynomials $g$ and $f_1,\dots,f_p$, all in $\Bbbk[x_1,\dots,x_n]$ for some field $\Bbbk$, we consider the problem of computing the critical points of the restriction of $g$ to the variety defined by $f_1=\cdots=f_p=0$. These are…

We consider extension of a closure system on a finite set S as a closure system on the same set S containing the given one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set…

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic…