Logarithmic-exponential series

✍ Scribed by Lou van den Dries; Angus Macintyre; David Marker

• Book ID: 104307110
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 379 KB
• Volume: 111
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(01)00035-5

✦ Synopsis

We extend the ÿeld of Laurent series over the reals in a canonical way to an ordered di erential ÿeld of "logarithmic-exponential series" (LE-series), which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the ÿeld of LE-series is a universal domain for ordered di erential algebra in Hardy ÿelds. We deÿne composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subÿelds of the ÿeld of LE-series are also considered.