# Identit\'es pond\'er\'ees Peirce-\'evanescentes

## TL;DR

This paper introduces procedures for constructing Peirce-evanescent identities in baric algebras, explores their properties using algebraic systems and rooted trees, and demonstrates their applicability to mutation algebras and spectrum analysis.

## Contribution

It provides new methods for generating Peirce-evanescent identities and links these identities to algebraic structures and spectra in baric algebras.

## Key findings

- Mutation algebras satisfy all Peirce-evanescent identities
- Any element of the field can be a Peirce spectrum of an algebra satisfying such identities
- Methods for generating identities are applicable to various algebraic cases

## Abstract

Peirce-evanescent baric identities are polynomial identities verified by baric algebras such that their Peirce polynomials are the null polynomial. In this paper procedures for constructing such homogeneous and non homogeneous identities are given. For this we define an algebraic system structure on the free commutative nonassociative algebra generated by a set T which provides for classes of baric algebras satisfying a given set of identities similar properties to those of the varieties of algebras. Rooted binary trees with labeled leaves are used to explain the Peirce polynomials. It is shown that the mutation algebras satisfy all Peirce-evanescent identities, it results from this that any part of the field K can be the Peirce spectrum of a K- algebra satisfying a Peirce-evanescent identity. We end by giving methods to obtain generators of homogeneous and non-homogeneous Peirce-evanescent identities that are applied in several univariate and multivariate cases.