Applications of Descartes' Rule of Signs

Let $\lambda$ be a positive proper fraction, $n$ a positive integer. The transcendental equation

has a single positive root. This increases monotonically to infinity as $n$ tends to infinity.

The function $x^{-5} \left(e^x - 1\right)^{-1}$ of the positive variable $x$ tends to $0$ for $x=0$ and for $x=+\infty$ and between these values has a maximum and no minimum.

Let $R(f)$ denote the number of reversals of sign of the function $f(x)$ in $(0,1)$ and $Z(K_n)$ the number of zeros of the polynomial

(see the introduction to problem 144 of Part Two) in $(0,1)$. Prove that

Let the radius of convergence of the power series $a_0+a_1x+a_2x^2+\cdots$ be $\geq 1$. The number of its zeros in the interval $0 < x < 1$ does not exceed the number of changes of sign of the sequence

Let $C$ be the number of changes of sign of the finite sequence

set

and assume that $P(1)=0$. Then $P(x)$ has at most $C+1$ positive zeros. Show also that the bound $C+1$ can be attained.

Let $\left(\dfrac {n}{p}\right)$ denote the Legendre (Jacobi) symbol, defined for $p$ an odd prime and $n$ any number not divisible by $p$ as $\left(\dfrac {n}{p}\right) = 1$ if $n$ is a quadratic residue of $p$ and $\left(\dfrac {n}{p}\right) = -1$ if $n$ is a quadratic non-residue of $p$ (Hardy--Wright, p. 68). The equation

has only one positive root, namely $x=1$.

The equation of degree $162$

has exactly $5$ positive roots which are all simple.

If the polynomial $a_0+a_1x+\cdots +a_nx^n$ has only real zeros, then $Z=C$.

Let $\nu_1, \nu_2, \ldots , \nu_n$ be integers, $0 \leq \nu_1 < \nu_2 < \cdots < \nu_n$, and $0 < \alpha_1 < \alpha_2 < \cdots < \alpha_n$. Then the determinant (a generalization of the Vandermonde determinant, which corresponds to the case $\nu_1=0, \nu_2=1, \ldots , \nu_n=n-1$)

Let $a_0 \neq 0$, $a_n \neq 0$, and assume that $2m$ consecutive coefficients of the polynomial $a_0+a_1x+\cdots +a_nx^n$ vanish, where $m$ is an integer, $m \geq 1$. Then the polynomial has at least $2m$ non-real zeros.

Let the polynomial $P(x)$ have only real zeros, and let $P(0)=1$ and $P(x)$ not a constant. Set

Then the polynomial $1+b_1x+\cdots +b_{2m}x^{2m}$ has only non-real zeros.

Let $m$ be an integer, $m \geq 1$, and

where the sum $\sum$ is taken over all sets of non-negative integers $l_1, l_2, \ldots , l_n$ for which

The homogeneous symmetric function $S(x_1, x_2, \ldots , x_n)$ of the $n$ real variables $x_1, x_2, \ldots , x_n$ is positive definite, i.e. $>0$ for all sets of values of $x_1, x_2, \ldots , x_n$ with the exception of the single set $x_1=0, x_2=0, \ldots , x_n=0$.

Let the polynomial $P(x) = x^n + \cdots$ have only positive zeros. In the power series expansion

all the coefficients $B_n, B_{n+1}, \ldots$ are positive.