相关论文: Computing Complex Dimension Faster and Determinist…
We extend the character sum method for the computation of densities in Artin primitive root problems developed by H. W. Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the…
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…
The study of sums of finite sets of integers has mostly concentrated on sets with small sumsets (Freiman's theorem and related work) and on sets with large sumsets (Sidon sets and $B_h$-sets). This paper considers the sets ${\mathcal…
Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency,…
The field of computational complexity is concerned both with the intrinsic hardness of computational problems and with the efficiency of algorithms to solve them. Given such a problem, normally one designs an algorithm to solve it and sets…
Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a…
For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…
In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new…
The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…
The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total…
A new algorithms for computing discrete logarithms on elliptic curves defined over finite fields is suggested. It is based on a new method to find zeroes of summation polynomials. In binary elliptic curves one is to solve a cubic system of…
In this work, following the discrete de Rham (DDR) approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the…
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
Hilbert's Nullstellensatz is a fundamental result in algebraic geometry that gives a necessary and sufficient condition for a finite collection of multivariate polynomials to have a common zero in an algebraically closed field. Associated…
We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum…
In this work we study the space complexity of computable real numbers represented by fast convergent Cauchy sequences. We show the existence of families of trascendental numbers which are logspace computable, as opposed to algebraic…