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.