Effective Differential Lüroth's Theorem

2012/02/28 by Lisi D'Alfonso, D'Alfonso, Lisi, Gabriela Jeronimo +3
Computer Science · Mathematics · #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Symbolic Computation (cs.SC) #cs.SC #math.AC

paper · pdf · doi:10.48550/arxiv.1202.6344

arxiv created 2013/07/01 · arxiv updated 2013/07/03

Abstract

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_n∈ F<u> generating a differential subfield G⊆ F<e u>. The differential Lüroth's theorem proved by Ritt in 1932 states that there exists v∈ \mathcal G such that G= F<v>. Here we prove that the total order and degree of a generator v are bounded by min_j \textrmord (v_j) and (nd(e+1)+1)^2e+1, respectively, where e:=max_j \textrmord (v_j) and d:=max_j \textrmdeg (v_j). 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.

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