| Title:
|
On realizability of sign patterns by real polynomials (English) |
| Author:
|
Kostov, Vladimir |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
853-874 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
real polynomial in one variable |
| Keyword:
|
sign pattern |
| Keyword:
|
Descartes' rule of signs |
| MSC:
|
26C10 |
| MSC:
|
30C15 |
| idZBL:
|
Zbl 06986977 |
| idMR:
|
MR3851896 |
| DOI:
|
10.21136/CMJ.2018.0163-17 |
| . |
| Date available:
|
2018-08-09T13:16:24Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147373 |
| . |
| Reference:
|
[1] Albouy, A., Fu, Y.: Some remarks about Descartes' rule of signs.Elem. Math. 69 (2014), 186-194. Zbl 1342.12002, MR 3272179, 10.4171/EM/262 |
| Reference:
|
[2] Anderson, B., Jackson, J., Sitharam, M.: Descartes' rule of signs revisited.Am. Math. Mon. 105 (1998), 447-451. Zbl 0913.12001, MR 1622513, 0913.12001 |
| Reference:
|
[3] Forsgård, J., Kostov, V. P., Shapiro, B.: Could René Descartes have known this?.Exp. Math. 24 (2015), 438-448. Zbl 1326.26027, MR 3383475, 10.1080/10586458.2015.1030051 |
| Reference:
|
[4] Grabiner, D. J.: Descartes' rule of signs: another construction.Am. Math. Mon. 106 (1999), 845-856. Zbl 0980.12001, MR 1732666, 10.2307/2589619 |
| Reference:
|
[5] Kostov, V. P.: Topics on Hyperbolic Polynomials in One Variable.Panoramas et Synthèses 33, Société Mathématique de France (SMF), Paris (2011). Zbl 1259.12001, MR 2952044 |
| Reference:
|
[6] Shapiro, B. Z., Khesin, B. A.: Swallowtails and Whitney umbrellas are homeomorphic.J. Algebr. Geom. 1 (1992), 549-560. Zbl 0790.57019, MR 1174901 |
| . |
