Some remarks on the non-real roots of polynomials

TL;DR

This paper investigates the real roots of a family of polynomials parameterized by t, revealing how their roots depend on the roots of the discriminant and the polynomial g(x), and constructs examples of totally complex reducible polynomials.

Contribution

It provides a detailed analysis of the real roots of polynomials of the form x^n + t g(x), relating root counts to the roots of the discriminant and polynomial g(x), and constructs totally complex reducible polynomials.

Findings

Number of real roots depends on the roots of the discriminant and g(x).

For large parameter ta, root counts follow specific parity-based rules.

Constructs examples of totally complex reducible polynomials.

Abstract

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of real roots of $g(x)=\sum_{k=0}^s t_{s-k} x^{s-k}$. For any $\xi > | \beta_m |$, if $n-s$ is odd then the number of real roots of $f(\xi, x)$ is $\gamma+1$, and if $n-s$ is even then the number of real roots of $f(\xi, x)$ is $\gamma$, $\gamma+2$ if $t_s>0$ or $t_s < 0$ respectively. A special case of the above result is constructing a family of totally complex polynomials which are reducible over $\mathbb Q$.