Positive root isolation for poly-powers by exclusion and differentiation

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, each contains exactly one positive root and together contain all, which can be easily refined 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, based on Gelfond-Schneider theorem, we present two complete isolation algorithms-exclusion and differentiation. For the latter, their completeness depends on Schanuel's conjecture. We implement the two methods and compare them in efficiency via a few examples. Finally the proposed methods are applied to the field of systems biology to show the practical usefulness.