Related papers: Choosing roots of polynomials smoothly, II
Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous…
In this paper, we consider the existence of a factorization of a monic, bounded motion polynomial. We prove existence of factorizations, possibly after multiplication with a real polynomial and provide algorithms for computing polynomial…
June Huh and Matthew Stevens conjectured that the Hilbert-Poincar\'e series of the Chow ring of any matroid is a polynomial with only real zeros. We prove this conjecture for the class of uniform matroids. We also prove that the Chow…
After reconsidering the theorem of continuity of the roots of a polynomial in terms of its coefficients in the deformation framework, we study the stability of the greater common divisor of two polynomials compared to perturbations on their…
In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is…
Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ are operational independently with probability $p \in[0,1]$. For fixed vertices $s$ and $t$, the \emph{two-terminal reliability} of $G$ is the probability…
In this paper we extend a previous investigation by us regarding an iterative construction of irreducible polynomials over finite fields of odd characteristic. In particular, we show how it is possible to iteratively construct irreducible…
This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present…
In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial…
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
We consider the set $\Pi ^*_d$ of monic polynomials $Q_d=x^d+\sum _{j=0}^{d-1}a_jx^j$, $x\in \mathbb{R}$, $a_j\in \mathbb{R}^*$, having $d$ distinct real roots, and its subsets defined by fixing the signs of the coefficients $a_j$. We show…
We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…
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…
We construct parametric families of (monic) reducible polynomials having two roots very close to each other.
Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or…
Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.
We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized…
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots)…
We study the real roots of the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlev\'e equation. It has been conjectured that the number of real roots of the nth…