Virtual roots of real polynomials (original) (raw)

Something You Always Wanted to Know About Real Polynomials (But Were Afraid to Ask)

arXiv: Classical Analysis and ODEs, 2017

The famous Descartes' rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomials in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration for generations of mathematicians. Trying to extend and sharpen this rule, we consider below the set of all real univariate polynomials of a given degree, a given collection of signs of their coefficients, and a given number of positive and negative roots. In spite of the elementary definition of the main object of our study, it is a non-trivial question for which sign patterns and numbers of positive and negative roots the corresponding set is non-empty. The main result of the present paper is a discovery of a new infinite family of non-realizable combinations of sign patterns and the numbers of positive and negative roots.

The Root Separation of Polynomials and Some Applications

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1995

PETKOVIC, M.; MIGNOTTE, M.; TRAJKOVIC, M. : Root Separation of Polynomials 55 1 I ZAMM . Z. angew. Math. Mech. 75 (1995) 7, 551 -561 PETKOVI~, M.; MIGNOTTE, M.; TRAJKOVIC, M.

Approximate Roots and the Hidden Geometry of Polynomial Coefficients

2020

The algebraic operation of approximate roots provides a geometric approximation of the zeros of a polynomial in the complex plane given conditions on their symmetry. A polynomial of degree n corresponds to a cluster of n zeros in the complex plane. The zero of the nth approximate root polynomial locates the gravitational center of this cluster. When the polynomial is of degree mn, with m clusters of n zeros, the centers of the clusters are no longer identified by the zeros of the nth approximate root polynomial in general. The approximation of the centers can be recovered given assumptions about the symmetric distribution of the zeros within each cluster, and given that m > n. Rouch´e’s theorem is used to extend this result to relax some of these conditions. This suggests an insight into the geometry of the distribution of zeros within the complex plane hidden within the coefficients of polynomials.

On the number of real roots of polynomials

Pacific Journal of Mathematics, 1982

A general theorem concerning the structure of a certain real algebraic curve is proved. Consequences of this theorem include major extensions of classical theorems of Pόlya and Schur on the reality of roots of polynomials.

The polynomial roots repartition and minimum roots separation

WSEAS Transactions on Mathematics, 2008

It is known that, if all the roots of a polynomial are real, they can be localised, using a set of intervals, which contain the arithmetic average of the roots. The aim of this paper is to present an original method for giving other distributions of the roots/ modules of the roots, on real axis, a method for evaluating and improving the "polynomial minimum root separation" results, a method for the complex polynomials and for polynomials having all roots real. We use the discriminant, Hadamard's inequality, Mahler's measure and new original inequalities. Also we will make some considerations about the cost for isolate the polynomial real roots. Our method is based on the successive splitting for the interval which contain all roots.

Polynomials, Sign Patterns and Descartes’ Rule

2019

The famous Descartes’ rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomial in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration for generations of mathematicians. Trying to extend and sharpen this rule, we consider below the set of all real univariate polynomials of a given degree, a given collection of signs of their coefficients, and given numbers of positive and negative roots. In spite of the elementary definition of the main object of our study, it is a non-trivial question for which sign patterns and numbers of positive and negative roots the corresponding set is non-empty. The main result of the present paper is a discovery of a new infinite family of non-realizable combinations of sign patterns and the numbers of positive and negative roots.

Roots of polynomials with positive coefficients

A necessary condition for stability of a finitedimensional linear time-invariant system is that all the coefficients of the characteristic equation are strictly positive. However, it is well-known that this condition is not sufficient, except for n less than 3. In this paper, we show that any polynomial that has positive coefficients cannot have roots on the nonnegative real axis. Conversely, if a polynomial has no roots on the positive real axis, a polynomial with positive coefficients can be found so that the product of the two polynomials also has positive coefficients. A simple upper bound for the degree of this multiplier polynomial is given. One application of the main result is that under a strict condition, it is possible to find a non-minimal realization of a given transfer function using only positive multipliers (except for the "minus" in the standard feedback comparator).

Forward stable computation of roots of real polynomials with only real distinct roots

2015

As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be chosen real. By using accurate forward stable algorithm for computing eigenvalues of real symmetric arrowhead matrices from (Jakovcevic Stor, Slapnicar, Barlow, 2015), we derive a forward stable algorithm for computation of roots of such polynomials in O(n2)O(n^2)O(n2) operations. The algorithm computes each root to almost full accuracy. In some cases, the algorithm invokes extended precision routines, but only in the non-iterative part. Our examples include numerically difficult problems, like the well-known Wilkinson's polynomials. Our algorithm compares favourably to other method for polynomial root-finding, like MPSolve or Newton's method.