Definition:Variation in Sign of Polynomial

Definition

$\pm a_n x^n \mathop \pm a_{n - 1} x^{n - 1} \mathop \pm \cdots \mathop \pm a_1 x \mathop \pm a_0 = 0$

where:

the coefficients $a_j \in \R_{\ge 0}$ are all non-negative constants

each of the $\pm$ signs can be either positive or negative.

A variation in sign of the coefficients of $\map P x$ is an instance of $j \in \set {0, 1, \ldots, n}$ in which a $\pm$ sign changes from positive to negative, or vice versa, as $j$ goes from $n$ to $0$, ignoring zero coefficients.

That is, it is a change of sign in the tuple $\tuple {a_n, a_{n - 1}, \ldots, a_0}$.

$x^5 + x^4 - 2 x^3 + x^2 - 1 = 0$

This has three variations in sign:

from $x^4$ to $-2 x^3$, where it goes from positive to negative

from $-2 x^3$ to $x^2$, where it goes from negative to positive

from $x^2$ to $-1$, where it goes from positive to negative.