Positive Root Isolation for Poly-Powers

Abstract

We consider a class of univariate real functions---poly-powers---that extend integer exponents to real algebraic exponents for polynomials. Our purpose is to isolate positive roots of such a function into disjoint intervals, which can be further easily computed up to any desired precision. To this end, we first classify poly-powers into simple and non-simple ones, depending on the number of linearly independent exponents. For the former, we present a complete isolation method based on Gelfond--Schneider theorem. For the latter, the completeness depends on Schanuel's conjecture. Finally experiential results demonstrate the effectivity of the proposed method.