Related papers: logcf: An Efficient Tool for Real Root Isolation
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(\log\log n)$.…
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…
We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence…
An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained…
We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…
Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in…
We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when…
This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called…
We present a FFT-based algorithm for the computation of a polynomial's coefficients from its roots, and apply it to obtain the coefficients of interpolation polynomials, to invert Vandermondians and to evaluate the symmetric functions of a…
We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order $1$. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign…
We study two important operations on polynomials defined over complete discrete valuation fields: Euclidean division and factorization. In particular, we design a simple and efficient algorithm for computing slope factorizations, based on…
Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have…
The reason why Cooley-Tukey Fast Fourier Transform (FFT) over $\mathbb{Q}$ can be efficiently implemented using complex roots of unity is that the cyclotomic extensions of the completion $\mathbb{R}$ of $\mathbb{Q}$ are at most quadratic,…
We propose an efficient algorithm for solving orthogonal canonical correlation analysis (OCCA) in the form of trace-fractional structure and orthogonal linear projections. Even though orthogonality has been widely used and proved to be a…
This paper proposes a convex optimization based method that either locates all real roots of a set of power flow equations or declares no real solution exists in the given area. In the proposed method, solving the power flow equations is…
We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).
We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of the roots of a $C^{n-1,1}$-curve of monic polynomials of degree $n$ is locally…
We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously…
Let $f(x)$ be a separable polynomial over a local field. Montes algorithm computes certain approximations to the different irreducible factors of $f(x)$, with strong arithmetic properties. In this paper we develop an algorithm to improve…