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Berarducci, A. and Mantova, V. (2018): Surreal numbers, derivations and transseries, Journal of the European Mathematical Society 20, pp. 339-390. arixv:1503.00315.

Berarducci, A. and Mantova, V. (2018): Surreal numbers, derivations and transseries, Journal of the European Mathematical Society 20, pp. 339-390. arixv:1503.00315.

Edit. 5/17/20.

Abstract: In (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $\mathbf{No}$, which includes $\mathbf{No}$ itself, extend to canonical exponential functions on their surcomplex counterparts. This uses the precursory result that trigonometric-exponential initial subfields of $\mathbf{No}$ and trigonometric ordered initial subfields of $\mathbf{No}$, more generally, admit \emph{canonical} sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $\mathbf{No}$, to the image of the canonical map of the ordered exponential field $\mathbb{T}$ of transseries into $\mathbf{No}$, which is shown to be initial, and to the ordered exponential fields $\mathbb{R}((\omega))^{EL}$ and $\mathbb{R}\langle\langle\omega\rangle \rangle$, which are likewise shown to be initial.

Edit. May 17, 2020.

Abstract: In (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $\mathbf{No}$, which includes $\mathbf{No}$ itself, extend to canonical exponential functions on their surcomplex counterparts. This uses the precursory result that trigonometric-exponential initial subfields of $\mathbf{No}$ and trigonometric ordered initial subfields of $\mathbf{No}$, more generally, admit canonical sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $\mathbf{No}$, to the image of the canonical map of the ordered exponential field $\mathbb{T}$ of transseries into $\mathbf{No}$, which is shown to be initial, and to the ordered exponential fields $\mathbb{R}((\omega))^{EL}$ and $\mathbb{R}\langle\langle\omega\rangle \rangle$, which are likewise shown to be initial.

Edit. 5/17/20.

The following recent paper by Elliot Kaplan and myself, adds further credence to the idea that the algebraico-tree-theoretic simplicity hierarchy is of critical importance to the surreals.

Surreal ordered exponential fields: arXiv:2002.07739

Edit. 5/17/20.

The following recent paper by Elliot Kaplan and myself adds further credence to the idea that the algebraico-tree-theoretic simplicity hierarchy is of critical importance to the surreals.

Surreal ordered exponential fields: (https://arxiv.org/abs/2002.07739)

Abstract: In (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $\mathbf{No}$, which includes $\mathbf{No}$ itself, extend to canonical exponential functions on their surcomplex counterparts. This uses the precursory result that trigonometric-exponential initial subfields of $\mathbf{No}$ and trigonometric ordered initial subfields of $\mathbf{No}$, more generally, admit \emph{canonical} sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $\mathbf{No}$, to the image of the canonical map of the ordered exponential field $\mathbb{T}$ of transseries into $\mathbf{No}$, which is shown to be initial, and to the ordered exponential fields $\mathbb{R}((\omega))^{EL}$ and $\mathbb{R}\langle\langle\omega\rangle \rangle$, which are likewise shown to be initial.