Related papers: Rationalizing roots: an algorithmic approach

We elaborate on the expansion of hypergeometric functions about rational parameters, where we focus mainly on the integer and half-integer case. The strategy and the basic steps of a recently developed algorithm for the expansion about…

We present a new algorithm for computing $m$-th roots over the finite field $\F_q$, where $q = p^n$, with $p$ a prime, and $m$ any positive integer. In the particular case $m=2$, the cost of the new algorithm is an expected $O(\M(n)\log (p)…

Suppose $A=\{a_1,\ldots,a_{n+2}\}\subset\mathbb{Z}^n$ has cardinality $n+2$, with all the coordinates of the $a_j$ having absolute value at most $d$, and the $a_j$ do not all lie in the same affine hyperplane. Suppose $F=(f_1,\ldots,f_n)$…

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…

Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric…

The reciprocal square root is an important computation for which many sophisticated algorithms exist (see for example \cite{Moroz,863046,863031} and the references therein). A common theme is the use of Newton's method to refine the…

A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals w.r.t. the space-time dimension $d$ is proposed. The relation between $d$ and $d-2$ dimensional integrals is given in terms of a…

In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the…

The systematic approach to solving the recurrence relations for multi-loop integrals is described. In particular, the criteria of their reducibility is suggested.

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently,…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in…

In this paper we present an algorithm to obtain the parameter planes of families of root-finding methods with several free critical points. The parameter planes show the joint behaviour of all critical points. This algorithm avoids the…

In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. We extend its application to the case of formal power series over a field of arbitrary characteristic and illustrate the proposed approach…

Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we…

Iterative methods for the simultaneous determination of all roots of an equation are dis-cussed. The multiplicities of the roots are assumed to be known in advance. The methods are proved to have a cubical rate of convergence. Numerical…

In this talk I review the connections between Feynman integrals and multiple polylogarithms. After an introductory section on loop integrals I discuss the Mellin-Barnes transformation and shuffle algebras. In a subsequent section multiple…

We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.

We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of…

We investigate the problem of factorizing a matrix into several sparse matrices and propose an algorithm for this under randomness and sparsity assumptions. This problem can be viewed as a simplification of the deep learning problem where…