On realizability of sign patterns by real polynomials

Vladimir Kostov

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 3, page 853-874
  • ISSN: 0011-4642

Abstract

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers ( p , n ) , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.

How to cite

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Kostov, Vladimir. "On realizability of sign patterns by real polynomials." Czechoslovak Mathematical Journal 68.3 (2018): 853-874. <http://eudml.org/doc/294785>.

@article{Kostov2018,
abstract = {The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p,n)$, chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials.},
author = {Kostov, Vladimir},
journal = {Czechoslovak Mathematical Journal},
keywords = {real polynomial in one variable; sign pattern; Descartes' rule of signs},
language = {eng},
number = {3},
pages = {853-874},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On realizability of sign patterns by real polynomials},
url = {http://eudml.org/doc/294785},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Kostov, Vladimir
TI - On realizability of sign patterns by real polynomials
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 853
EP - 874
AB - The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p,n)$, chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials.
LA - eng
KW - real polynomial in one variable; sign pattern; Descartes' rule of signs
UR - http://eudml.org/doc/294785
ER -

References

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  1. Albouy, A., Fu, Y., 10.4171/EM/262, Elem. Math. 69 (2014), 186-194. (2014) Zbl1342.12002MR3272179DOI10.4171/EM/262
  2. Anderson, B., Jackson, J., Sitharam, M., 0913.12001, Am. Math. Mon. 105 (1998), 447-451. (1998) Zbl0913.12001MR1622513DOI0913.12001
  3. Forsgård, J., Kostov, V. P., Shapiro, B., 10.1080/10586458.2015.1030051, Exp. Math. 24 (2015), 438-448. (2015) Zbl1326.26027MR3383475DOI10.1080/10586458.2015.1030051
  4. Grabiner, D. J., 10.2307/2589619, Am. Math. Mon. 106 (1999), 845-856. (1999) Zbl0980.12001MR1732666DOI10.2307/2589619
  5. Kostov, V. P., Topics on Hyperbolic Polynomials in One Variable, Panoramas et Synthèses 33, Société Mathématique de France (SMF), Paris (2011). (2011) Zbl1259.12001MR2952044
  6. Shapiro, B. Z., Khesin, B. A., Swallowtails and Whitney umbrellas are homeomorphic, J. Algebr. Geom. 1 (1992), 549-560. (1992) Zbl0790.57019MR1174901

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