Effective differential Lüroth's theorem
This paper focuses on effectivity aspects of the Lüroth's theorem in differential fields. Let F be an ordinary differential field of characteristic 0 and F〈u〉 be the field of differential rational functions generated by a single indeterminate u. Let be given non-constant rational functions v1,v...
| Authors: | , , |
|---|---|
| Format: | article |
| Status: | Published version |
| Publication Date: | 2014 |
| Country: | Argentina |
| Institution: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repository: | CONICET Digital (CONICET) |
| Language: | English |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/93864 |
| Online Access: | http://hdl.handle.net/11336/93864 |
| Access Level: | Open access |
| Keyword: | 12H05 12Y05 DIFFERENTIAL ALGEBRA DIFFERENTIATION INDEX LÜROTH'S THEOREM https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Summary: | This paper focuses on effectivity aspects of the Lüroth's theorem in differential fields. Let F be an ordinary differential field of characteristic 0 and F〈u〉 be the field of differential rational functions generated by a single indeterminate u. Let be given non-constant rational functions v1,vn∈F〈u〉 generating a differential subfield G⊆F〈u〉. The differential Lüroth's theorem proved by Ritt in 1932 states that there exists v∈G such that G=F〈v〉. Here we prove that the total order and degree of a generator v are bounded by minjord(vj) and (n d(e +1) +1)2e +1, respectively, where e:=maxjord(vj) and d:=maxjdeg(vj). As a byproduct, our techniques enable us to compute a Lüroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables. |
|---|