Related papers: Improved complexity bounds for real root isolation…

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of…

We design polynomial size, constant depth (namely, $\mathsf{AC}^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, B\'ezout coefficients,…

This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…

A method is given for finding roots of a one-variable function using Taylor's expansion of that function and fractional derivative calculated at a suitable tangent point without using Newton's method, but is regarded as a variant of Halley…

We present three new, practical algorithms for polynomials in $\mathbb{Z}[x]$: one to test if a polynomial is cyclotomic, one to determine which cyclotomic polynomials are factors, and one to determine whether the given polynomial is…

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials…

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…

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $\alpha$ of $f$ in the algebraic closure of $F$,…

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…

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 consider polynomials of degree $d$ with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of polynomials at a fixed point off the real line. There are two explicit families of…

We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on…

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying…

Let L be any number field or $\mathfrak{p}$-adic field and consider F:=(f_1,...,f_k) where f_i is in L[x_1,...,x_n]\{0} for all i and there are exactly m distinct exponent vectors appearing in f_1,...,f_k. We prove that F has no more than…

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erd\H{o}s-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than…

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…

We extend Robust Optimization to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use Robust…

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construc-tion uses a single linear differential form defined from the…