A known Lemma on the largest root of a polynomial and its derivatives? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:06:21Z http://mathoverflow.net/feeds/question/72614 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72614/a-known-lemma-on-the-largest-root-of-a-polynomial-and-its-derivatives A known Lemma on the largest root of a polynomial and its derivatives? Mike 2011-08-10T19:21:19Z 2011-08-12T07:50:18Z <p>Greetings,</p> <p>I am currently working on a paper that involves an upper bound of the largest root of a polynomial. With the help of the Mean Value Theorem, I believe a colleague and I have proved the following: if f^(n)(p) > 0 (the n-th derivative evaluated at "p", for n=0,1,2..d, where d is the degree of the polynomial), then p>r, where r is the largest root of the polynomial.</p> <p>The only question I have is the following: is this well-known or has been cited/proven elsewhere? It just seems basic enough that someone in the past has referenced it in a paper/Lemma. Though we could be mistaken and have a flaw in our own proof...</p> <p>thanks,</p> <p>Mike</p> http://mathoverflow.net/questions/72614/a-known-lemma-on-the-largest-root-of-a-polynomial-and-its-derivatives/72623#72623 Answer by Bruno for A known Lemma on the largest root of a polynomial and its derivatives? Bruno 2011-08-10T21:00:15Z 2011-08-12T07:50:18Z <p>For me, it is a folklore result for people interested in Descartes' rule of signs, even though I am not aware of any paper stating it explicitly. It follows from Descartes' rule of signs.</p> <p><strong>Theorem</strong> (Descartes' rule of signs). The number of positive roots of a real polynomial is not larger than the number of sign changes in the sequence of its coefficients.</p> <p>You can prove the following generalization: </p> <p><strong>Lemma.</strong> Let $P$ be a degree-$d$ polynomial and $r\in\mathbb R$. The number of sign changes in the sequence of the $P^{(n)}(r)$ for $n=0$ to $d$ is an upper bound on the number of roots of P larger than $r$. </p> <p><em>Proof.</em> Let $Q(x)=P(x+r)$. Then $Q^{(n)}(0)/n!$ is the coefficient of $x^n$ in $Q$. Therefore, the number of signs changes in the sequence $(P^{(n)}(r))_n$ equals the number of sign changes in the sequence of coefficients of $Q$. By Descartes' rule of signs, this number of sign changes upper bounds the number of positive roots of $Q$. $\square$ </p> <p><em>Remark.</em> It is a consequence of Rolle's Theorem since Rolle's Theorem is used to prove Descartes' rule of signs.</p>