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...

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Autores principales: D'Alfonso, L., Jeronimo, G., Solernó, P.
Formato: JOUR
Materias:
12H05
12Y05
Differential algebra
Differentiation index
Lüroth's theorem
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00218693_v406_n_p1_DAlfonso
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_00218693_v406_n_p1_DAlfonso
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spelling todo:paper_00218693_v406_n_p1_DAlfonso2023-10-03T14:21:32Z Effective differential Lüroth's theorem D'Alfonso, L. Jeronimo, G. Solernó, P. 12H05 12Y05 Differential algebra Differentiation index 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,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. © 2014 Elsevier Inc. Fil:Jeronimo, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Solernó, P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00218693_v406_n_p1_DAlfonso
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic 12H05
12Y05
Differential algebra
Differentiation index
Lüroth's theorem
spellingShingle 12H05
12Y05
Differential algebra
Differentiation index
Lüroth's theorem
D'Alfonso, L.
Jeronimo, G.
Solernó, P.
Effective differential Lüroth's theorem
topic_facet 12H05
12Y05
Differential algebra
Differentiation index
Lüroth's theorem
description 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. © 2014 Elsevier Inc.
format JOUR
author D'Alfonso, L.
Jeronimo, G.
Solernó, P.
author_facet D'Alfonso, L.
Jeronimo, G.
Solernó, P.
author_sort D'Alfonso, L.
title Effective differential Lüroth's theorem
title_short Effective differential Lüroth's theorem
title_full Effective differential Lüroth's theorem
title_fullStr Effective differential Lüroth's theorem
title_full_unstemmed Effective differential Lüroth's theorem
title_sort effective differential lüroth's theorem
url http://hdl.handle.net/20.500.12110/paper_00218693_v406_n_p1_DAlfonso
work_keys_str_mv AT dalfonsol effectivedifferentiallurothstheorem
AT jeronimog effectivedifferentiallurothstheorem
AT solernop effectivedifferentiallurothstheorem
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