The Lagrangian remainder of Taylors series, distinguishes $mathcal{O}(f(x))$ time complexities to polynomials or not

The purpose of this letter is to investigate the time complexity consequences of the truncated Taylor series, known as Taylor Polynomials \cite{bakas2019taylor,Katsoprinakis2011,Nestoridis2011}. In particular, it is demonstrated that the examination of the $\mathbf{P=NP}$ equality, is associated with the determination of whether the $n^{th}$ derivative of a particular solution is bounded or not. Accordingly, in some cases, this is not true, and hence in general.

💡 Research Summary

The manuscript proposes a novel criterion for distinguishing whether a time‑complexity expression (O(f(x))) represents a polynomial‑time algorithm (i.e., belongs to class P) by examining the boundedness of the ((n+1)^{\text{st}}) derivative of the underlying function (f). Starting from the Taylor expansion of an analytic function (f) around a point (x_{0}),

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