Exponential polynomial inequalities and monomial sum inequalities in p -Newton sequences

Charles R. Johnson; Carlos Marijuán; Miriam Pisonero; Michael Yeh

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 793-819
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Johnson, Charles R., et al. "Exponential polynomial inequalities and monomial sum inequalities in $\rm p$-Newton sequences." Czechoslovak Mathematical Journal 66.3 (2016): 793-819. <http://eudml.org/doc/286846>.

@article{Johnson2016,
abstract = {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.},
author = {Johnson, Charles R., Marijuán, Carlos, Pisonero, Miriam, Yeh, Michael},
journal = {Czechoslovak Mathematical Journal},
keywords = {exponential polynomial; Newton inequality; Newton coefficients; p-Newton sequence},
language = {eng},
number = {3},
pages = {793-819},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Exponential polynomial inequalities and monomial sum inequalities in $\rm p$-Newton sequences},
url = {http://eudml.org/doc/286846},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Johnson, Charles R.
AU - Marijuán, Carlos
AU - Pisonero, Miriam
AU - Yeh, Michael
TI - Exponential polynomial inequalities and monomial sum inequalities in $\rm p$-Newton sequences
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 793
EP - 819
AB - 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.
LA - eng
KW - exponential polynomial; Newton inequality; Newton coefficients; p-Newton sequence
UR - http://eudml.org/doc/286846
ER -

References

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  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. (2013) Zbl1305.15022MR3090453
  2. Johnson, C. R., Marijuán, C., Pisonero, M., Matrices and spectra satisfying the Newton inequalities, Linear Algebra Appl. 430 (2009), 3030-3046. (2009) Zbl1189.15009MR2517856
  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. (2011) MR3051171
  4. Newton, I., Arithmetica Universalis: Sive de Compositione et Resolutione Arithmetica Liber, William Whiston London (1707). (1707) 
  5. Wang, X., 10.2307/4145072, Am. Math. Mon. 111 (2004), 525-526. (2004) DOI10.2307/4145072

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