Levi-Civita field

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

If ${\displaystyle f=\sum \limits _{q\in \mathbb {Q} }f_{q}\varepsilon ^{q}}$ and ${\displaystyle g=\sum \limits _{q\in \mathbb {Q} }g_{q}\varepsilon ^{q}}$ are two Levi-Civita series, then

• their sum ${\displaystyle f+g}$ is the pointwise sum ${\displaystyle f+g:=\sum \limits _{q\in \mathbb {Q} }(f_{q}+g_{q})\varepsilon ^{q}}$.
• their product ${\displaystyle fg}$ is the Cauchy product ${\displaystyle fg:=\sum \limits _{q\in \mathbb {Q} }\sum \limits _{a+b=q}(f_{a}g_{b})\varepsilon ^{q}}$.

(One can check that the support of this series is left-finite and that for each of its elements ${\displaystyle q}$, the set ${\displaystyle \{(a,b)\in \mathbb {Q} \times \mathbb {Q}$ :\ a+b=q\wedge f_{a}\neq 0\wedge g_{b}\neq 0\}} is finite, so the product is well defined.)

• the relation ${\displaystyle 0<f}$ holds if ${\displaystyle f\neq 0}$ (i.e. ${\displaystyle f}$ has non-empty support) and the least non-zero coefficient of ${\displaystyle f}$ is strictly positive.

Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of ${\displaystyle \mathbb {R} }$ where the series ${\displaystyle \varepsilon }$ is a positive infinitesimal.

The Levi-Civita field is real-closed, meaning that it can be algebraically closed by adjoining an imaginary unit (i), 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.[1]

The Levi-Civita field is also Cauchy complete, meaning that relativizing the ${\displaystyle \forall \exists \forall }$ 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 ${\displaystyle \mathbb {R} }$, and the value group is ${\displaystyle (\mathbb {Q} ,+)}$. 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 ${\displaystyle (\mathbb {Q} ,+)}$ is a proper immediate extension, containing series such as ${\displaystyle 1+\varepsilon ^{1/2}+\varepsilon ^{2/3}+\varepsilon ^{3/4}+\varepsilon ^{4/5}+\cdots }$ which are not in the Levi-Civita field.

The Levi-Civita field is the Cauchy-completion of the field ${\displaystyle \mathbb {P} }$ of Puiseux series over the field of real numbers, that is, it is a dense extension of ${\displaystyle \mathbb {P} }$ without proper dense extension. Here is a list of some of its notable proper subfields and its proper ordered field extensions:

• A web-based calculator for Levi-Civita numbers