Cubic equations with 2 Roots in the interval $[-1, 1]$

TL;DR

The paper addresses the problem of determining when exactly two roots of a monic cubic $P(x)=x^3+ a x^2+ b x+ c$ lie in the closed interval $[-1,1]$. It develops a discriminant-guided, geometry-backed framework that combines the discriminant $D_3$ with coefficient-invariants $A$, $B$, $A_T$, $B_T$, and $E$, and performs a case analysis across the parameter $c$—including special cusp behavior and plane intersections—to yield explicit, region-based criteria for two-root configurations. It also extends the analysis to cases with zero/one/three roots in the interval and to polynomials with complex-conjugate pairs, supported by numerical checks and an application to rigid-body tops (nutation) exemplified by the Lagrange top. The approach provides a transparent, visualization-driven method for root-counting in a fixed interval, with potential applicability to other interval-constrained polynomial problems in physics and applied mathematics.

Abstract

The conditions for cubic equations, to have 3 real roots and 2 of the roots lie in the closed interval $[-1, 1]$ are given. These conditions are visualized. This question arises in physics in e.g. the theory of tops.

Figures (9)

• Figure 1: $c = 1$, the 2 components of $D_3 = 0$, the 2 parabola $P_a, P_b$ The cusp is located in the dark grey shaded lens. The cusps for all $c$ lie on the red parabola $P_C$.
• Figure 2: $c = 0$, i.e. a root $0$ and roots of the quadratic polynomial $x^2 + a x + b$, in red the number of roots of the quadratic in the interval $[-1, 1]$, the green lines represent polynomials with 1 root in the interval the blue curve is the parabola for the discriminant $D_2 = 0$.
• Figure 3: $c = 1 / 4$, roots of the cubic polynomial $x^3 + a x^2 + b x + c$, in red the number of roots in the interval $[-1, 1]$. The point $A = A_{I \, 2}$ is the intersection of the line $B$ with the discriminant $D_3$.
• Figure 4: $c = 1 / 4$, the cusp, roots of the cubic polynomial $x^3 + a x^2 + b x + c$, in red the number of roots in the interval $[-1, 1]$. The point $A = A_{I \, 1}$ is the intersection of the line $B$ with the discriminant $D_3$.
• Figure 5: $c = 1$, roots of the cubic polynomial $x^3 + a x^2 + b x + c$, in red the number of roots in the interval $[-1, 1]$.