Related papers: Univariate real root isolation in an extension fie…
The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree $N$ polynomial chosen…
We formulate the Root Extraction problem in finite Abelian $p$-groups and then extend it to generic finite Abelian groups. We provide algorithms to solve them. We also give the bounds on the number of group operations required for these…
We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem…
We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative…
The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive'…
We present a symbolic-numeric method to refine an approximate isolated singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, ..., \hat{x}_{n})$ of a polynomial system $F=\{f_1, ..., f_n\}$ when the Jacobian matrix of $F$ evaluated at…
We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain $\Omega$ in $\mathbb R^m$ admit a parameterization by functions of bounded variation uniformly with respect to the…
We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct…
Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for…
We consider the problem of deterministically factoring a univariate polynomial over a finite field under the assumption of the Extended Riemann Hypothesis (ERH). This work builds upon the line of approach first explored by Gao in $2001$.…
In this paper we describe how to improve the performance of the symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case. By…
In this paper, we present several new linearizations of a quadratic binary optimization problem (QBOP), primarily using the method of aggregations. Although aggregations were studied in the past in the context of solving system of…
A polynomial is real-rooted if all of its roots are real. For every polynomial $f(t) \in {\mathbf R}[t]$, the Hermite-Sylvester theorem associates a quadratic form $\Phi_2$ such that $f(t)$ is real-rooted if and only if $\Phi_2$ is positive…
We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m=3 we can approximate within eps all the roots of f in the interval [0,R] using just O(log(D)log(Dlog(R/eps)))…
Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors.…
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…
The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structure that a set of real roots is endowed with (mutual position…
Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our new near-optimal root-finders approximate all zeros of a polynomial p almost as fast as one accesses its coefficients with…