Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis

In the framework of Polynomial Eigenvalue Problems (PEPs), most matrix polynomials arising in applications are structured (namely, (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve PEPs is by means linearizations. frequently used linearizations belong general constructions, valid for all a fixed degree, known as companion It well known, however, that it not possible construct preserve any previous structures even degree. This motivates search more forms, particular $\ell$-ifications. this paper, we present, first time, family (generalized) $\ell$-ifications these structures, degree $k=(2d+1)\ell$. We also show how sparse within family. Finally, prove there no quadratifications quartic polynomials.