Levi-Civita field

• 7\varepsilon is an infinitesimal that is greater than \varepsilon, but less than every positive real number. • \varepsilon^2 is less than \varepsilon, and is also less than r\varepsilon for any positive real r. • 1+\varepsilon differs infinitesimally from 1. • \varepsilon^{1/2} is greater than \varepsilon and even greater than r\varepsilon for any positive real r, but \varepsilon^{1/2} is still less than every positive real number. • 1/\varepsilon is greater than any real number. • 1+\varepsilon+\varepsilon^2/2+\cdots+\varepsilon^n/n!+\cdots is interpreted as e^\varepsilon, which differs infinitesimally from 1. • 1+\varepsilon + 2\varepsilon^2 + \cdots + n!\varepsilon^n + \cdots is a valid member of the field, because the series is to be constructed formally, without any consideration of convergence. It differs infinitesimally from 1 and is smaller than 1+2\varepsilon. ==Definition of the field operations and positive cone==

If a=\textstyle\sum \limits_{q \in \mathbb{Q}}a_q \varepsilon^q and b=\textstyle\sum \limits_{q \in \mathbb{Q}}b_q \varepsilon^q are two Levi-Civita series, then • their sum a+b is the pointwise sum a+b:=\textstyle\sum \limits_{q \in \mathbb{Q}}(a_q+b_q) \varepsilon^q. • their product ab is the Cauchy product ab:=\textstyle\sum\limits_{q \in \mathbb{Q}}\big(\!\sum \limits_{r+s=q} a_r b_s \big)\varepsilon^q. One can check that for every q\in\mathbb{Q} the set :\{(r,s) \in \mathbb{Q}\times\mathbb{Q}: \ r+s=q,\ a_r \neq 0,\ b_s \neq 0\} is finite, so that all the products are well-defined, and that the resulting series defines a valid Levi-Civita series. Moreover, • their quotient is defined by polynomial long division. Division of two finite series may result in an infinite series, analogous to infinite decimal expansions. • the relation 0 holds if a\neq 0 (i.e. at least one coefficient of a is non-zero) and the least non-zero coefficient of a is strictly positive. Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of \mathbb{R} where the series \varepsilon is a positive infinitesimal. ==Properties and applications==

The Levi-Civita field is real-closed, meaning that it can be algebraically closed by adjoining an imaginary unit, or by letting the coefficients be complex. It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented using floating point. It is the basis of automatic differentiation, a way to perform differentiation in cases that are intractable by symbolic differentiation or finite-difference methods. The Levi-Civita field is also Cauchy complete, meaning that relativizing the \forall \exists\forall definitions of Cauchy sequence and convergent sequence to sequences of Levi-Civita series, each Cauchy sequence in the field converges. Equivalently, it has no proper dense ordered field extension. As an ordered field, it has a natural valuation given by the rational exponent corresponding to the first non zero coefficient of a Levi-Civita series. The valuation ring is that of series bounded by real numbers, the residue field is \mathbb{R}, and the value group is (\mathbb{Q},+). The resulting valued field is Henselian (being real closed with a convex valuation ring) but not spherically complete. Indeed, the field of Hahn series with real coefficients and value group (\mathbb{Q},+) is a proper immediate extension, containing series such as 1+\varepsilon^{1/2}+\varepsilon^{2/3}+\varepsilon^{3/4}+\varepsilon^{4/5}+\cdots which are not in the Levi-Civita field. ==Relations to other ordered fields==

The Levi-Civita field is the Cauchy-completion of the field \mathbb{P} of Puiseux series over the field of real numbers, that is, it is a dense extension of \mathbb{P} without proper dense extension. Here is a list of some of its notable proper subfields and its proper ordered field extensions: Notable subfields • The field \mathbb{R} of real numbers. • The field \mathbb{R}(\varepsilon) of fractions of real polynomials (rational functions) with infinitesimal positive indeterminate \varepsilon. • The field \mathbb{R}((\varepsilon)) of formal Laurent series over \mathbb{R}. • The field \mathbb{P} of Puiseux series over \mathbb{R}. Notable extensions • The field \mathbb{R}\varepsilon^{\mathbb{Q}} of Hahn series with real coefficients and rational exponents. • The field \mathbb{T}^{LE} of logarithmic-exponential transseries. • The field \mathbf{No}(\varepsilon_0) of surreal numbers with birthdate below the ordinal number \varepsilon_0. • Fields of hyperreal numbers constructed as ultrapowers of \mathbb{R} modulo a free ultrafilter on \mathbb{N} (although here the embeddings are not canonical). ==References==