Superalgebras and Algebras with Involution: Classifying Cubic Codimension Sequences

A $\phi$-algebra is either a superalgebra or an algebra with involution. In this paper, we study ${\mathrm{T}}^{\phi }$-ideals associated with unital $\phi$-algebras whose $\phi$-codimension sequence exhibits cubic polynomial growth. As a consequence, we obtain a complete classification of all $\phi$-codimension sequences of cubic growth for unital $\phi$-algebras. Furthermore, we explicitly determine a minimal-degree multilinear generator for every ${\mathrm{T}}^{\phi }$-ideal associated with unital $\phi$-algebras whose $\phi$-codimension growth is at most quadratic.