On multivariate minimal polynomials

Abstract

Given compact sets E and F in [Copf ]ⁿ (n [ges ] 1) related by F = q⁻¹(E) where q is a polynomial map, we are interested in the general problem of comparing minimal polynomials for E with minimal polynomials for F. Let α be an n-multi-index of length d. We define the classes of polynomials ℙ(α) := z^α + [Copf ]_d−1[z] and [Pscr ](α) := {p: p(z) = z^α + [sum ]_{β[pr ]α}a_βz^β} where [pr ] denotes the usual graded lexicographic order. Polynomials in ℙ(α) or in [Pscr ](α) of least deviation from zero on E (with respect to the supremum norm) are called minimal polynomials for E. We prove that if q is a simple (i.e. qˆ_i(z) = z^m_i) polynomial mapping of degree m and if p is minimal polynomial for E then p ∘ q is a minimal polynomial for F = q⁻¹(E) and, using some algebraic machinery, we can also construct minimal polynomials for E from minimal polynomials for F. The result seems to be new even in the one-dimensional case.