Related papers: The Newton-Puiseux algorithm and effective algebra…
We prove a theorem on algebraic osculation and we apply our result to the Computer Algebra problem of polynomial factorization. We consider X a smooth completion of the complex plane and D an effective divisor supported on the boundary of…
Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
This work represents a natural coalescence of two important lines of work: learning mixtures of Gaussians and algorithmic robust statistics. In particular we give the first provably robust algorithm for learning mixtures of any constant…
This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation,…
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,…
It is described how the Hermite-Pad\'e polynomials corresponding to an algebraic approximant for a power series may be used to predict coefficients of the power series that have not been used to compute the Hermite-Pad\'e polynomials. A…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
The problem of effective equations is reviewed and discussed. Starting from the classical Langevin equation, we show how it can be generalized to Hamiltonian systems with non-standard kinetic terms. A numerical method for inferring…
The ideas behind the concept of algebraic ("integration-by-parts") algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the…
An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This…
We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are…
Recent results in the theory and application of Newton-Puiseux expansions, i.e. fractional power series solutions of equations, suggest further developments within a more abstract algebraic-geometric framework, involving in particular the…
Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the…
We present module theory and linear maps as a powerful generalised and computationally efficient framework for the relational data model, which underpins today's relational database systems. Based on universal constructions of modules we…
Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial…
The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by…
Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…