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- W1818695112 abstract "This paper focuses on effectivity aspects of the Luroth's theorem in differential fields. Let $mathcal{F}$ be an ordinary differential field of characteristic 0 and $mathcal{F} $ be the field of differential rational functions generated by a single indeterminate $u$. Let be given non constant rational functions $v_1,...,v_nin mathcal{F} $ generating a differential subfield $mathcal{G}subseteq mathcal{F} $. The differential Luroth's theorem proved by Ritt in 1932 states that there exists $vin mathcal G$ such that $mathcal{G}= mathcal{F} $. Here we prove that the total order and degree of a generator $v$ are bounded by $min_j textrm{ord} (v_j)$ and $(nd(e+1)+1)^{2e+1}$, respectively, where $e:=max_j textrm{ord} (v_j)$ and $d:=max_j textrm{deg} (v_j)$. As a byproduct, our techniques enable us to compute a Luroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables." @default.
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- W1818695112 date "2012-02-28" @default.
- W1818695112 modified "2023-09-27" @default.
- W1818695112 title "Effective Differential Luroth's Theorem" @default.
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