A note on generalized companion pencils in the monomial basis

Abstract

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Teran et al. (Linear Algebra Appl 459:264–333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360–365, 2016), and Ma–Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621–625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Teran and Hernando (INdAM Series, Springer, Berlin, pp 157–179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencil.