Commutative algebras satisfying univariate identities with vanishing Peirce polynomial

We introduce and study $(2,3)$-palintropic algebras, a class of commutative algebras defined by the identity $(x^{3})^2 - (x^{2})^3 = 0$.

This specific relation is the simplest generator of the $2$-dimensional space of minimal-degree evanescent identities in degree $6$, and encompasses several well-studied structures, including Jordan and medial algebras.

The primary motivation for investigating these algebras lies in their trivial Peirce polynomials, which removes a priori restrictions on the spectrum of the multiplication operator associated with an idempotent.

In this paper, we review and further develop the theory of Peirce operators, Peirce polynomials, and second-order linearizations.

We demonstrate that despite the triviality of the Peirce polynomial, any idempotent $c$ admits well-behaved, explicit fusion rules for multiplication between its $\lambda$-Peirce spaces for $\lambda \neq \tfrac{1}{2}$.

Furthermore, we prove that multiplication by such an idempotent always constitutes an algebra homomorphism.

Finally, we present concrete examples of $(2,3)$-palintropic algebras and provide applications of these algebraic structures to commutative polynomial maps.