A non-realization theorem in the context of Descartes' rule of signs
Annual of Sofia University "St. Kliment Ohridski'', Faculty of Mathematics and Informatics vol. 106 (2019) 25-51 For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Des...
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Zusammenfassung: | Annual of Sofia University "St. Kliment Ohridski'', Faculty of
Mathematics and Informatics vol. 106 (2019) 25-51 For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with
$c$ sign changes and $p$ sign preservations in the sequence of its coefficients
($c+p=d$), Descartes' rule of signs says that $P$ has $pos\leq c$ positive and
$neg\leq p$ negative roots, where $pos\equiv c($\, mod $2)$ and $neg\equiv
p($\, mod $2)$. 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 $(pos,
neg)$ satisfying these conditions there exists a polynomial $P$ with exactly
$pos$ positive and $neg$ negative roots (all of them simple); that is, all
these cases are realizable. This is not true for $d\geq 4$, yet for $4\leq
d\leq 8$ (for these degrees the exhaustive answer to the question of
realizability is known) in all nonrealizable cases either $pos=0$ or $neg=0$.
It was conjectured that this is the case for any $d\geq 4$. For $d=9$, we show
a counterexample to this conjecture: for the sign pattern
$(+,-,-,-,-,+,+,+,+,-)$ and the pair $(1,6)$ there exists no polynomial with
$1$ positive, $6$ negative simple roots and a complex conjugate pairs and, up
to equivalence, this is the only case for $d=9$. |
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DOI: | 10.48550/arxiv.1911.12255 |