On the Structure of Radical Towers and their Parametrizations

Abstract

Radical parametrizations describe algebraic varieties by means of nested radicals of polynomials, and their theory relies on three varieties associated with a parametrization, namely, the incidence, tower and radical varieties. In this paper we revisit this framework from a computational point of view and show that it is governed by the radical tower rather than by the particular parametrization. We prove that the tower variety depends only on the tower and that the incidence variety is always irreducible, so that no prime decomposition is needed. Moreover, we show how to replace a tower by an isomorphic one whose tower variety is defined by the equations of the tower, so that it can be computed without elimination. We introduce universal radical parametrizations and we use them to settle, under a natural hypothesis on the coefficients, an open case in the transformation of differential equations with radical coefficients into equations with rational coefficients. We show that radical subtowers form a lattice, which yields minimal radical parametrizations, and we give algorithms to compute them. We also prove that dominance between towers induces a unique dominant rational map between their tower varieties, through which unirationality is transferred. Finally, we study towers over subfields of $\mathbb{C}$ and, in the real case, relate the real points of the tower variety to the real domains of the associated parametrizations.