Research
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A Geometric Classification of the Depressed Cubic Family
Discriminant, Multiplicity, and Bifurcations in Parameter Space
Rafael Fuentes Rangel
Expository mathematical note· Jul 20, 2026 · Version 1.1
A study of the real two-parameter family f(x) = x³ + ax + b, showing how the discriminant partitions parameter space by the number and multiplicity of real roots. The bifurcation boundary is a semicubical cusp, and it corresponds exactly to the appearance of a multiple root.
- cubic-polynomials
- discriminants
- multiple-roots
- bifurcations
- parameter-space
- singularity-theory
- algebra
- calculus