Related papers: A review on computational aspects of polynomial am…
This article is a survey on the topic of polynomial amoebas. We review results of papers written on the topic with an emphasis on its computational aspects. Polynomial amoebas have numerous applications in various domains of mathematics and…
We present algorithms for computation and visualization of amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide method and an algorithm for the computation…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
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 are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on…
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
Every polyhedral cone can be described either by its facets or by its extreme rays. Computation of one description from the other is a problem that can be very complex, i.e. one encounter the combinatorial explosion. We present here several…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C++ type languages are discussed together with means that provide for computations with an…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, P\'olya…
Computational problems can be classified according to their algorithmic complexity, which is defined based on how the resources needed to solve the problem, e.g. the execution time, scale with the problem size. Many problems in…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include…
We apply methods of machine-learning, such as neural networks, manifold learning and image processing, in order to study 2-dimensional amoebae in algebraic geometry and string theory. With the help of embedding manifold projection, we…