Liouville’s theorem on integration in finite terms for D∞,SL2, and Weierstrass field extensions

• Published in: Archiv der Mathematik 2023, Vol.121 (4), p.371-383
• Main Authors: Kumbhakar, Partha, Srinivasan, Varadharaj R.
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Subgroups; Theorems
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-023-01907-5

Let k be a differential field of characteristic zero and the field of constants C of k be an algebraically closed field. Let E be a differential field extension of k having C as its field of constants and that E = E m ⊇ E m - 1 ⊇ ⋯ ⊇ E 1 ⊇ E 0 = k , where E i is either an elementary extension of E i - 1 or E i = E i - 1 ( t i , t i ′ ) and t i is Weierstrassian (in the sense of Kolchin (Amer. J. Math. 75(4):753–824, 1953)) over E i - 1 or E i is a Picard–Vessiot extension of E i - 1 having a differential Galois group isomorphic to either the special linear group SL 2 ( C ) or the infinite dihedral subgroup D ∞ of SL 2 ( C ) . In this article, we prove that Liouville’s theorem on integration in finite terms (Rosenlicht in Pac J Math 24(1):153–161, 1968, Theorem) holds for E . That is, if η ∈ E and η ′ ∈ k , then there is a positive integer n and for i = 1 , 2 , ⋯ , n , there are elements c i ∈ C , u i ∈ k \ { 0 } , and v ∈ k such that η ′ = ∑ i = 1 n c i u i ′ u i + v ′ .