# Bounds for Absolute Positiveness of Multivariate Polynomials

@article{Hong1998BoundsFA, title={Bounds for Absolute Positiveness of Multivariate Polynomials}, author={Hoon Hong}, journal={J. Symb. Comput.}, year={1998}, volume={25}, pages={571-585} }

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.

## 36 Citations

Bounds on absolute positiveness of multivariate polynomials

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2010

Faster algorithms for computing Hong's bound on absolute positiveness

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2010

On the Quality of Some Root-Bounds

- Mathematics, Computer ScienceMACIS
- 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

- MathematicsMultidimens. Syst. Signal Process.
- 2001

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

- Mathematics, Computer ScienceCASC 2015
- 2015

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

- MathematicsCASC
- 2007

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

- Computer Science, MathematicsCASC
- 2016

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

- MathematicsJ. Univers. Comput. Sci.
- 2009

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

- Computer Science, MathematicsISSAC '07
- 2007

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

- Mathematics
- 2008

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…

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