On the metaphysics of ${\mathbb{F}}_1$

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 ${\mathbb{F}}_1$.

Dedicated to Yuri Manin, 14 Figures