Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
real polynomial in one variable; sign pattern; Descartes' rule of signs
real polynomial in one variable; sign pattern; Descartes' rule of signs
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.
References:
[1] Albouy, A., Fu, Y.: Some remarks about Descartes' rule of signs. Elem. Math. 69 (2014), 186-194. DOI 10.4171/EM/262 | MR 3272179 | Zbl 1342.12002
[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.
[3] Forsgård, J., Kostov, V. P., Shapiro, B. Z.: Could René Descartes have known this?. Exp. Math. 24 (2015), 438-448. DOI 10.1080/10586458.2015.1030051 | MR 3383475 | Zbl 1326.26027
[4] Forsgård, J., Kostov, V. P., Shapiro, B. Z.: Corrigendum:"Could René Descartes have known this?". (to appear) in Exp. Math. DOI 10.1080/10586458.2017.1417775 | MR 3383475
[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.
[6] Gauss, C. F.: Beweis eines algebraischen Lehrsatzes. J. Reine Angew. Math. 3 (1828), 1-4 German. DOI 10.1515/crll.1828.3.1 | MR 1577673 | Zbl 003.0089cj
[7] Grabiner, D. J.: Descartes' rule of signs: Another construction. Am. Math. Mon. 106 (1999), 854-856. DOI 10.2307/2589619 | MR 1732666 | Zbl 0980.12001
[8] Kostov, V. P.: On realizability of sign patterns by real polynomials. (to appear) in Czech. Math. J. DOI 10.21136/CMJ.2018.0163-17 | MR 3851896
[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
Search
Partner of
