Determinantal representations in umbral calculus

In this paper, we construct a duality functor for affine rational curves given by polynomials in echelon form. This provides an explicit way to realize dual bases for the classical polarity pairing, and also yields known results on determinantal representations of Appell sequences of polynomials. We also provide a finite-difference version of these constructions, that allows to build approximate polarity pairings as well as a corresponding Grace theorem with a controlled optimal error term.