Levi-Civita field

In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Its members can be constructed as formal series of the form

$\sum_{q\in\mathbb{Q}} a_q\varepsilon^q ,$

where $a_q\,$ are real numbers, $\mathbb{Q}$ is the set of rational numbers, and $\varepsilon$ is to be interpreted as a positive infinitesimal. The support of a, i.e., the set of indices of the nonvanishing coefficients $\{q\in\mathbb{Q}:a_q\neq 0\},$ must be a left-finite set: for any member of $\mathbb{Q}$, there are only finitely many members of the set less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to dictionary ordering of the list of coefficients, which is equivalent to the assumption that $\varepsilon$ is an infinitesimal.

The real numbers are embedded in this field as series in which all of the coefficients vanish except $a_0\,$.

Examples

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

Extensions and applications

The field 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]

Hahn series (with real coefficients and value group $\mathbb{Q}$) are a larger field which relaxes the condition on the support $\{q\in\mathbb{Q}:a_q\neq 0\}$ of being left finite to that of being well-ordered (i.e., admitting no infinite decreasing sequence): this gives a meaning to 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.

References

1. Khodr Shamseddine, "Analysis on the Levi-Civita Field: A Brief Overview", http://www.physics.umanitoba.ca/~khodr/Publications/RS-Overview-offprints.pdf

External links

• A web-based calculator for Levi-Civita numbers