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Keywords:
exponential polynomial; Newton inequality; Newton coefficients; p-Newton sequence
exponential polynomial; Newton inequality; Newton coefficients; p-Newton sequence
Summary:
We consider inequalities between sums of monomials that hold for all p-Newton sequences. This continues recent work in which inequalities between sums of two, two-term monomials were combinatorially characterized (via the indices involved). Our focus is on the case of sums of three, two-term monomials, but this is very much more complicated. We develop and use a theory of exponential polynomial inequalities to give a sufficient condition for general monomial sum inequalities, and use the sufficient condition in two ways. The sufficient condition is necessary in the case of sums of two monomials but is not known if it is for sums of more. A complete description of the desired inequalities is given for Newton sequences of less than 5 terms.
References:
[1] Johnson, C. R., Marijuán, C., Pisonero, M.: Inequalities for linear combinations of monomials in p-Newton sequences. Linear Algebra Appl. 439 (2013), 2038-2056. MR 3090453 | Zbl 1305.15022
[2] Johnson, C. R., Marijuán, C., Pisonero, M.: Matrices and spectra satisfying the Newton inequalities. Linear Algebra Appl. 430 (2009), 3030-3046. MR 2517856 | Zbl 1189.15009
[3] Johnson, C. R., Marijuán, C., Pisonero, M., Walch, O.: Monomials inequalities for Newton coefficients and determinantal inequalities for p-Newton matrices. Trends in Mathematics, Notions of Positivity and the Geometry of Polynomials Springer, Basel Brändén, Petter et al. (2011), 275-282. MR 3051171
[4] Newton, I.: Arithmetica Universalis: Sive de Compositione et Resolutione Arithmetica Liber. William Whiston London (1707).
[5] Wang, X.: A simple proof of Descartes's rule of signs. Am. Math. Mon. 111 (2004), 525-526. DOI 10.2307/4145072
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