# Article

 Title: Exponential polynomial inequalities and monomial sum inequalities in $\rm p$-Newton sequences (English) Author: Johnson, Charles R. Author: Marijuán, Carlos Author: Pisonero, Miriam Author: Yeh, Michael Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 66 Issue: 3 Year: 2016 Pages: 793-819 Summary lang: English . Category: math . 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. (English) Keyword: exponential polynomial Keyword: Newton inequality Keyword: Newton coefficients Keyword: p-Newton sequence MSC: 11C20 MSC: 15A15 MSC: 15A18 MSC: 15A45 idZBL: Zbl 06644034 idMR: MR3556868 DOI: 10.1007/s10587-016-0293-7 . Date available: 2016-10-01T15:25:09Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/145872 . Reference: [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. Zbl 1305.15022, MR 3090453 Reference: [2] Johnson, C. R., Marijuán, C., Pisonero, M.: Matrices and spectra satisfying the Newton inequalities.Linear Algebra Appl. 430 (2009), 3030-3046. Zbl 1189.15009, MR 2517856 Reference: [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 Reference: [4] Newton, I.: Arithmetica Universalis: Sive de Compositione et Resolutione Arithmetica Liber.William Whiston London (1707). Reference: [5] Wang, X.: A simple proof of Descartes's rule of signs.Am. Math. Mon. 111 (2004), 525-526. 10.2307/4145072 .

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