Archibald, Beal, Birkhoff, Blichfeldt, B. H. Camp, Cole, Crum, Dantzig, Dodd, Douglas, Fine, Fiske, Fite, M. C. Foster, Philip Franklin, Fry, Gafafer, Gill, Gronwall, Hazlett, Hebbert, E. R. Hedrick, Hille, Huntington, Joffe, Kellogg, Kircher, Kuhn, Lamson, MacDuffee, Mathewson, Meder, Mirick, Molina, Mullins, Northcott, Oglesby, Osgood, Pell, Pfeiffer, Raynor, Reddick, R. G. D. Richardson, Rit...

This paper focuses on effectivity aspects of the Lüroth’s theorem in differential fields. Let F be a 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 subfield G ⊆ F〈u〉. The differential Lüroth’s theorem proved by Ritt in 1932 state...

We study the following questions: (1) What are all solutions to f ◦f̂ = g◦ĝ with f, g, f̂ , ĝ ∈ C(X) being complex rational functions? (2) For which rational functions f(X) and g(X) with rational coefficients does the equation f(a) = g(b) have infinitely many solutions with a, b ∈ Q? We utilize various algebraic, geometric and analytic results in order to resolve both (1) and a variant of (2) in ...

Classical methods to study and solve systems of polynomial equations are based on numerous avatars of Gröbner (standard) basis algorithms or Riquier-Janet type methods (Ritt-Wu’s algorithm). All these methods use implicitly but deeply the dense or sparse representation of multivariate polynomials, which is the computer science counterpart of the expansion of these mathematical objects on the mo...

We study the following questions: (1) What are all solutions to f ◦ f̂ = g ◦ ĝ in complex rational functions f, g ∈ C(X) and meromorphic functions f̂ , ĝ on the complex plane? (2) For which rational functions f(X) and g(X) with coefficients in an algebraic number field K does the equation f(a) = g(b) have infinitely many solutions with a, b ∈ K? We utilize various algebraic, geometric and analyti...