In 2009 a new theory in the Complex analysis, unifying the old concept of elementary functions with Ordinary Differential Equations (ODEs), Automatic Differentiation (AD), and analytic continuation, was presented . In it, two competing definitions of the vector elementary vs. scalar (or stand-alone) elementary functions were introduced, so that the question about their equivalency immediately surfaced up. The gap in the Unifying View was the Conjecture, claiming that a system of m first-order explicit polynomial ODEs may be converted into one n-order rational ODE with a nonzero denominator at the given point. Also earlier, in 2007 the similar question emerged in connection with a new type of special points where the function is holomorphic, but its scalar elementariness is violated . Such a function can satisfy no rational n-order ODE regular at this point. Is the similar statement true also for systems of rational ODEs? This question reduced to the same Conjecture, posed in both papers, remaining unsolved since 2008. Finally, in this paper the Conjecture is proved closing the gap in the Unifying View and finalizing this theory. The Conjecture was proved thanks to its reformulation into terms of algebra, and collaboration with algebraists George Bergman and Alexander Givental, exemplifying a case of a successful interdisciplinary synergy.
Mathematics Subject Classification. 34A09; 34A34; 34A12.