Abstract
In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of \mathbb{S}[\mu_{n,+}] -polynomials and relate this concept to the known notion of number systems. The Riemann–Roch theorem for the ring \mathbb{Z} of the integers that we obtained recently uses the understanding of \mathbb{Z} as a ring of polynomials \mathbb{S}[X] in one variable over the absolute base \mathbb{S} , where 1+1=X+X^{2} . The absolute base \mathbb{S} (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious \mathbb{F}_{1} .
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