Bounds for Absolute Positiveness of Multivariate Polynomials

  title={Bounds for Absolute Positiveness of Multivariate Polynomials},
  author={Hoon Hong},
  journal={J. Symb. Comput.},
  • H. Hong
  • Published 1 May 1998
  • Mathematics
  • J. Symb. Comput.
A multivariate polynomialP(x1, �,xn) with real coefficients is said to beabsolutely positivefrom a real numberBiff it and all of its non-zero partial derivatives of every order are positive forx1,�,xn�B. We call suchBaboundfor the absolute positiveness ofP. This paper provides a simple formula for computing such bounds. We also prove that the resulting bounds are guaranteed to be close to the optimal ones. 
Bounds on absolute positiveness of multivariate polynomials
Faster algorithms for computing Hong's bound on absolute positiveness
On the Quality of Some Root-Bounds
  • P. Batra
  • Mathematics, Computer Science
  • 2015
Knowing results are applied to show which are the salient features of the Lagrange real root-bound as well as the related bound by Fujiwara which are carried over to the bounds by Kioustelidis and Hong.
Enclosure of the Zero Set of Polynomials in Several Complex Variables
The zero set of one general multivariate polynomial is enclosed by unions and intersections of simple unbounded sets. Sets in which multivariate real polynomials exhibit constant sign or stay
A New Polynomial Bound and Its Efficiency
A new bound for absolute positiveness of univariate polynomials with real coefficients is proposed and its efficiency with respect to known bounds is discussed and compared with the threshold ofabsolute positiveness.
Bounds for Real Roots and Applications to Orthogonal Polynomials
New inequalities on the real roots of a univariate polynomial with real coefficients are obtained and estimates for the largest positive root are derived, which is a key step for real root isolation.
A Lower Bound for Computing Lagrange's Real Root Bound
The tradeoff between improving the quality of bounds on absolute positiveness and their computational complexity is explored and it is found that this is optimal in the real RAM model.
Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials
This paper reviews the existing linear and quadratic complexity bounds on the values of the positive roots of polynomials and their impact on the per- formance of the Vincent-Akritas-Strzebonski (VAS) continued fractions method and finds that VAS(lmq) is 40% faster than the original version VAS (cauchy).
Complexity of real root isolation using continued fractions
This paper derives the first polynomial worst case bound on the continued fraction algorithm: for a square-free integer poynomial of degree n and coefficients of bit-length L, the bit-complexity of the continued fractions algorithm is Õ(n7L2), using a bound by Hong to compute the floor of the smallest positive root of a polynometric.
Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots
In this paper we compare four implementations of the Vincent-Akritas- Strzebonski Continued Fractions (VAS-CF) real root isolation method using four different (two linear and two quadratic


An efficient algorithm for infallible polynomial complex root isolation
Although the algorithm is not designed for efficient refinement of isolating rectangles, it does nevertheless refine all c]f these rectangles to width 10-50 in less than 4 minutes.
Polynomials and Polynomial Inequalities
Chaptern 1 Introduction and Basic Properties.- 2 Some Special Polynomials.- 3 Chebyshev and Descartes Systems.- 4 Denseness Questions.- 5 Basic Inequalities.- 6 Inequalities in Muntz Spaces.-
Polynomial real root isolation by differentiation
A new algorithm is described, for the isolation of the real roots of a real polynomial, relying on Rolle's theorem and a tangent construction to decide whether an interval contains two roots or none.
Note on the roots of algebraic equations
(Read before the American Mathematical Society at Chicago, April 10, 1914.) 1. LANDAU* has established certain interesting inequalities concerning the least root of a class of algebraic equations,
An elementary double inequality for the roots of an algebraic equation having greatest absolute value
LET there be given an algebraic equation of the nth degree written x + CnAaix + Cnt2a2X + h an = 0, where (7n,i, Cn$, • • • denote binomial coefficients. Let X\9 X<L, • • • 9 xn denote its roots, and
Algorithms for exact polynomial root calculation
This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses
Algorithms for polynomial real root isolation
The improved algorithms presented in this thesis significantly reduce the computing time of Collins' cylindrical algebraic decomposition (CAD) based quantifier elimination (QE) algorithm.
Polynomial real root isolation using Descarte's rule of signs
Uspensky's 1948 book on the theory of equations presents an algorithm, based on Descartes' rule of signs, for isolating the real roots of a squarefree polynomial with real coefficients, which proves to be a strong competitor of the recently discovered algorithm of Collins and Loos.
Proving Polynomials Positive
Some observations on and several improvements of a method (developed by BenCherifa/Lescanne) for performing polynomial orderings proofs for proving the termination of term rewriting systems are provided.