| Title:
|
Polynomials, sign patterns and Descartes' rule of signs (English) |
| Author:
|
Kostov, Vladimir Petrov |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
144 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
39-67 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
By Descartes' rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has ${\rm pos}\leq c$ positive and $\neg \leq p$ negative roots, where ${\rm pos}\equiv c\pmod 2$ and $\neg \equiv p\pmod 2$. For $1\leq d\leq 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $({\rm pos}, {\rm neg})$ satisfying these conditions there exists a polynomial $P$ with exactly ${\rm pos}$ positive and exactly $\neg $ negative roots (all of them simple). For $d\geq 4$ this is not so. It was observed that for $4\leq d\leq 8$, in all nonrealizable cases either ${\rm pos}=0$ or ${\rm neg}=0$. It was conjectured that this is the case for any $d\geq 4$. We show a counterexample to this conjecture for $d=11$. Namely, we prove that for the sign pattern $(+,-,-,-,-,-,+,+,+,+,+,-)$ and the pair $(1,8)$ there exists no polynomial with $1$ positive, $8$ negative simple roots and a complex conjugate pair. (English) |
| Keyword:
|
real polynomial in one variable |
| Keyword:
|
sign pattern |
| Keyword:
|
Descartes' rule of signs |
| MSC:
|
26C10 |
| MSC:
|
30C15 |
| idZBL:
|
Zbl 07088835 |
| idMR:
|
MR3934197 |
| DOI:
|
10.21136/MB.2018.0091-17 |
| . |
| Date available:
|
2019-03-21T12:31:05Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147638 |
| . |
| 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] Cajori, F.: A history of the arithmetical methods of approximation to the roots of numerical equations of one unknown quantity.Colorado College Publication, Science Series 12 (1910), 171-215, 217-287 \99999JFM99999 42.0057.02. |
| Reference:
|
[3] Forsgård, J., Kostov, V. P., Shapiro, B. Z.: Could René Descartes have known this?.Exp. Math. 24 (2015), 438-448. Zbl 1326.26027, MR 3383475, 10.1080/10586458.2015.1030051 |
| Reference:
|
[4] Forsgård, J., Kostov, V. P., Shapiro, B. Z.: Corrigendum:"Could René Descartes have known this?".(to appear) in Exp. Math. MR 3383475, 10.1080/10586458.2017.1417775 |
| Reference:
|
[5] Fourier, J.: Sur l'usage du théorème de Descartes dans la recherche des limites des racines.Bulletin des sciences par la Société Philomatique de Paris (1820), 156-165, 181-187 Oeuvres de Fourier publiées par les soins de M. Gaston Darboux sous les auspices du ministère de l'instruction publique. Tome II. Mémoires publiés dans divers recueils Gauthier-Villars, Paris 1890 291-309\kern0pt French \99999JFM99999 22.0021.01. |
| Reference:
|
[6] Gauss, C. F.: Beweis eines algebraischen Lehrsatzes.J. Reine Angew. Math. 3 (1828), 1-4 German. Zbl 003.0089cj, MR 1577673, 10.1515/crll.1828.3.1 |
| Reference:
|
[7] Grabiner, D. J.: Descartes' rule of signs: Another construction.Am. Math. Mon. 106 (1999), 854-856. Zbl 0980.12001, MR 1732666, 10.2307/2589619 |
| Reference:
|
[8] Kostov, V. P.: On realizability of sign patterns by real polynomials.(to appear) in Czech. Math. J. MR 3851896, 10.21136/CMJ.2018.0163-17 |
| Reference:
|
[9] Kostov, V. P., Shapiro, B.: Something you always wanted to know about real polynomials (but were afraid to ask).Avaible at https://arxiv.org/abs/1703.04436. |
| . |
