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Positive root isolation for poly-powers by exclusion and differentiation. (English) Zbl 1378.68200
Summary: We consider a class of univariate real functions – $$\mathsf{poly-powers}$$ – 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 $$\mathsf{poly-powers}$$ 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.

##### MSC:
 68W30 Symbolic computation and algebraic computation 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 12Y05 Computational aspects of field theory and polynomials (MSC2010) 92C42 Systems biology, networks