Bounds for Absolute Positiveness of Multivariate Polynomials

A multivariate polynomial P (x1, . . . , xn) with real coefficients is said to be absolutely positive from a real number B iff it and all of its non-zero partial derivatives of every order are positive for x1, . . . , xn ≥ B. We also call such B a bound for the absolute positiveness of P . The main goal of this paper is to devise a “nice” formula for computing a bound of a given polynomial. The initial motivation arose while studying several partial methods for testing positiveness of multivariate polynomials (Ben-Cherif and Lescanne, 1987; Dershowitz, 1987; Steinbach, 1992; Steinbach, 1994; Giesl, 1995). We found that these partial methods are in fact complete methods for testing absolute positiveness (Hong and Jakus, 1996). Since then, we have also realized that most previously known formulas for univariate root bounds (Cauchy, 1829; Birkhoff, 1914; Carmichael and Mason, 1914; Fujiwara, 1915; Kelleher, 1916; Kuniyeda, 1916; Cohn, 1922; Montel, 1932; Tôya, 1933; Berwald, 1934; Marden, 1949; Johnson, 1991) in fact bounds for absolute positiveness (thus, a bound not only for the polynomial, but also for all its non-zero derivatives). Indeed, from Lucas’ theorem (Lucas, 1874) one can conclude, in the univariate case, that any complex root bound, when used as a bound for real roots, is also a bound for absolute positiveness (see Section 5). Thus, we believe that the notion of absoluteness positiveness deserves to be investigated. Not all multivariate polynomials have bounds for absolute positiveness. Thus, first we need to have a method for checking the existence of bounds. An efficient method is given in Hong and Jakus (1996), and we use it in this paper.