TY - GEN
T1 - An additive decomposition in logarithmic towers and beyond
AU - Du, Hao
AU - Guo, Jing
AU - Li, Ziming
AU - Wong, Elaine
N1 - Publisher Copyright:
© 2020 ACM.
PY - 2020/7/20
Y1 - 2020/7/20
N2 - We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in a certain kind of primitive tower which we call S-primitive, as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower is integrable in the tower if and only if the remainder is equal to zero. The additive decomposition is achieved by viewing our towers not as a traditional chain of extension fields, but rather as a direct sum of certain subrings. Furthermore, we can determine whether or not a function in an S-primitive tower has an elementary integral without the need to deal with differential equations explicitly. We also show that any logarithmic tower can be embedded into a particular extension where we can further decompose the given function. The extension is constructed using only differential field operations without introducing any new constants.
AB - We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in a certain kind of primitive tower which we call S-primitive, as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower is integrable in the tower if and only if the remainder is equal to zero. The additive decomposition is achieved by viewing our towers not as a traditional chain of extension fields, but rather as a direct sum of certain subrings. Furthermore, we can determine whether or not a function in an S-primitive tower has an elementary integral without the need to deal with differential equations explicitly. We also show that any logarithmic tower can be embedded into a particular extension where we can further decompose the given function. The extension is constructed using only differential field operations without introducing any new constants.
KW - additive decomposition
KW - elementary integral
KW - logarithmic tower
KW - primitive tower
KW - symbolic integration
UR - http://www.scopus.com/inward/record.url?scp=85090359525&partnerID=8YFLogxK
U2 - 10.1145/3373207.3404025
DO - 10.1145/3373207.3404025
M3 - Conference contribution
AN - SCOPUS:85090359525
T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
SP - 146
EP - 153
BT - ISSAC 2020 - Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation
A2 - Mantzaflaris, Angelos
PB - Association for Computing Machinery
T2 - 45th International Symposium on Symbolic and Algebraic Computation, ISSAC 2020
Y2 - 20 July 2020 through 23 July 2020
ER -