A Geometric Classification of the Depressed Cubic Family

This is an expository mathematical note. The results are classical; the contribution is the organization, geometric interpretation, and presentation of the analysis.

Overview

A family of functions depending on parameters is one of the most accessible settings for a real mathematical investigation. Instead of studying one object, the question becomes how a property changes as the parameters move, and parameter space itself becomes the object of study: it splits into regions where the behavior is qualitatively the same, separated by a boundary where something degenerates.

The depressed cubic

$$f_{a,b}(x) = x^3 + ax + b, \qquad (a,b) \in \mathbb{R}^2,$$

is a compact model for this kind of investigation. Depressed means the quadratic term has been removed; every monic cubic can be brought to this form by translating the variable, so nothing is lost by studying it in this reduced shape. Despite its simplicity, the family already shows multiple roots, regions of constant real-root behavior, and a discriminant curve with a genuine singularity, a cusp.

The classification problem is simple to state: for which $(a,b)$ does $f_{a,b}$ have one real root, and for which does it have three? The answer turns out to be governed entirely by one quantity.

Main result

The sign of the discriminant

$$\Delta(a,b) = -4a^3 - 27b^2$$

completely determines the real-root structure of $f_{a,b}$.

$\Delta > 0$: three distinct real roots. $\Delta = 0$: a multiple root. $\Delta < 0$: exactly one real root (and a complex-conjugate pair).

In words: as $(a,b)$ crosses the curve $\Delta = 0$, two real roots collide and disappear into a complex-conjugate pair, or vice versa. That curve is the entire subject of what follows.

Critical points and global shape

The derivative is $f'_{a,b}(x) = 3x^2 + a$, so the shape of the graph depends only on the sign of $a$:

• $a > 0$: $f'_{a,b} > 0$ everywhere, so $f_{a,b}$ is strictly increasing and crosses zero exactly once.
• $a = 0$: $f'_{a,b}$ has a single degenerate zero at $x = 0$; the function is still increasing overall.
• $a < 0$: $f'_{a,b}$ has two distinct zeros, $x_\pm = \pm\sqrt{-a/3}$, giving a local maximum at $x_-$ and a local minimum at $x_+$.

The two critical points are what make three real roots possible at all. A monic cubic goes from $-\infty$ to $+\infty$, so it always crosses zero at least once; a second and third crossing can only appear if the graph first rises to a positive local maximum and then dips back down to a negative local minimum. Writing $s = \sqrt{-a/3}$, the critical values are

$$f_{a,b}(-s) = b + 2s^3, \qquad f_{a,b}(s) = b - 2s^3,$$

and three distinct real roots occur exactly when $-2s^3 < b < 2s^3$: the local maximum is positive and the local minimum is negative.

Discriminant and multiplicity

A repeated root is a point where the graph is tangent to the axis, not just crossing it. That means a multiple root $r$ must satisfy both

$$r^3 + ar + b = 0 \qquad \text{and} \qquad 3r^2 + a = 0$$

simultaneously: the function vanishes and its derivative vanishes at the same point. Solving the second equation for $a$ and substituting into the first gives

$$a = -3r^2, \qquad b = 2r^3.$$

Substituting these into $\Delta(a,b) = -4a^3 - 27b^2$ gives $\Delta = 0$ identically, for every $r$. So every multiple root traces out a curve in $(a,b)$-space, and $\Delta = 0$ is precisely that curve.

Geometry of the bifurcation curve

The bifurcation boundary is the algebraic curve

$$\mathcal{D} = \{(a,b) \in \mathbb{R}^2 : 4a^3 + 27b^2 = 0\},$$

and the substitution above gives it a clean polynomial parametrization:

$$\sigma(t) = (-3t^2,\ 2t^3), \qquad t \in \mathbb{R}.$$

Along this curve, the polynomial factors explicitly as a perfect square times a linear term:

$$f_{-3t^2,\,2t^3}(x) = (x - t)^2 (x + 2t).$$

The point $t = 0$ is special. There, $\sigma(0) = (0,0)$, and checking the partial derivatives of $G(a,b) = 4a^3 + 27b^2$ shows that the origin is the only point where the curve fails to be smooth: $G_a = 12a^2$ and $G_b = 54b$ both vanish only at $(a,b) = (0,0)$. Geometrically, the two branches of the curve meet at the origin with a shared tangent line rather than crossing transversally, the defining picture of a semicubical cusp.

Semicubical cusp in the a-b parameter plane. Points inside the cusp correspond to three distinct real roots, while points outside correspond to one real root. A draggable point shows the classification for any (a, b): drag it, or focus the figure and use the arrow keys.

a =

−2

b =

0

Δ =

32

roots:

Three distinct real roots

Root classification

Putting the pieces together gives a complete classification. Away from $a = 0$, write $a = -3s^2$; then $\Delta(a,b) = 27(2s^3 - b)(2s^3 + b)$, which is positive exactly on the three-root interval found above.

Condition	Shape of the graph	Real roots
$\Delta > 0$	positive local max, negative local min	three simple roots
$\Delta = 0$, $(a,b) \neq (0,0)$	tangent to the axis at one point	one double root, one simple root
$(a,b) = (0,0)$	degenerate horizontal inflection	one triple root
$\Delta < 0$	crosses the axis once	one simple root

When $a \geq 0$ the function is increasing everywhere, so $\Delta \leq 0$ automatically and equality holds only at the origin; the interesting case is entirely contained in $a < 0$, where the cusp lives.

Representative examples

Three real roots. For $(a,b) = (-3,0)$,

$$f_{-3,0}(x) = x^3 - 3x = x(x - \sqrt3)(x + \sqrt3), \qquad \Delta(-3,0) = 108 > 0.$$

Graph of f(x) = x^3 - 3x, crossing the x-axis at three distinct points: minus the square root of 3, zero, and the square root of 3.

A double root. For $(a,b) = (-3,2)$,

$$f_{-3,2}(x) = x^3 - 3x + 2 = (x-1)^2(x+2), \qquad \Delta(-3,2) = 0.$$

Graph of f(x) = x^3 - 3x + 2, factored as (x - 1) squared times (x + 2). It is tangent to the x-axis at the double root x = 1, and crosses it once more at the simple root x = -2.

One real root. For $(a,b) = (1,1)$, the polynomial $x^3 + x + 1$ is strictly increasing, and $\Delta(1,1) = -31 < 0$.

Graph of f(x) = x^3 + x + 1, a strictly increasing function that crosses the x-axis exactly once, near x = -0.68.

Further directions

The same method — locate the critical points, impose a degeneracy condition to derive the exceptional locus, then classify the connected components of its complement — extends to richer families. A natural next case is the quintic $g_{a,b,c}(x) = x^5 + ax^3 + bx + c$, whose parameter space is three-dimensional and whose discriminant surface has a more intricate singular geometry. Polynomial maps in two variables offer another direction, where the Jacobian determinant takes over the role of the ordinary derivative and questions of local and global invertibility arise.

Conclusion

The depressed cubic family gives a complete, visually transparent classification. The discriminant $\Delta = -4a^3 - 27b^2$ records both multiplicity and the number of real roots, while the locus $\Delta = 0$ is a cusp parametrized by $(-3t^2, 2t^3)$. It is a small example, but it shows how calculus, algebra, and elementary singularity theory settle the same question from three different directions at once.

Citing this note

Rafael Fuentes Rangel. “A Geometric Classification of the Depressed Cubic Family: Discriminant, Multiplicity, and Bifurcations in Parameter Space.” Expository mathematical note, version 1.1, July 20, 2026.

References

1. V. I. Arnold, Catastrophe Theory, 3rd ed., Springer, 1992.
2. D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, 4th ed., Springer, 2015.
3. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, 1994.
4. S. Lang, Algebra, 3rd ed., Springer, 2002.

@article{fuentesrangel2026depressedcubic,
  author = {Rafael Fuentes Rangel},
  title = {A Geometric Classification of the Depressed Cubic Family: Discriminant, Multiplicity, and Bifurcations in Parameter Space},
  year = {2026},
  month = {July},
  note = {Expository mathematical note, version 1.1},
  url = {https://rafablockdev.github.io/research/depressed-cubic-family/}
}