Related papers: Newton iteration, conditioning and zero counting

Continuous amortization is a technique for computing the complexity of algorithms, and it was first presented by the author in Burr, Krahmer, & Yap (2009). Continuous amortization can result in simpler and more straight-forward complexity…

We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to…

We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

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.…

We consider polynomially and rationally parameterized curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex…

We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the…

These notes are based on lecture courses I gave to third year mathematics students at Cambridge. They could form a basis of an elementary one--term lecture course on integrable systems covering the Arnold-Liouville theorem, inverse…

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are…

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…

We explain how to encode an algebraic series by finite data and how to do effective arithmetic on the level of these encodings. The reasoning is based on the Newton-Puiseux algorithm and an effective equality test for algebraic series.…

We study whether a Large Language Model can learn the deterministic sequence of trees generated by the iterated prime factorization of the natural numbers. Each integer is mapped into a rooted planar tree and the resulting sequence $…

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time…

Let I be an arbitrary ideal in C[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal…

The recent progresses in Machine Learning opened the door to actual applications of learning algorithms but also to new research directions both in the field of Machine Learning directly and, at the edges with other disciplines. The case…

A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification…

In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605, 2008) we analyzed a numerical algorithm for computing the number of real zeros of a…

The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic…

We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations…