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Nevanlinna Theory, Normal Families, and Algebraic Differential Equations pp 137–172Cite as

Algebraic Differential Equations

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Abstract

One of the most difficult problems in the theory of Algebraic Differential Equations is to decide whether or not the solutions are meromorphic in the plane. In case this question has been answered satisfactorily, which by experience requires particular strategies adapted to the equations under consideration, there remain several major problems to be solved: to determine the Nevanlinna functions, the distribution of zeros and poles, zero- and pole-free regions, asymptotic expansions on pole-free regions, and solutions deviating from the ‘generic’ case. This program will be pursued in the subsequent sections on linear, Riccati, and implicit first-order differential equations.

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Notes

  1. 1.

    Let f be a bounded holomorphic function on some unbounded domain D. By definition, the cluster set of f as z → ∞ on D is the set of all limits lim n → ∞ f(z n ) with z n → ∞ in D. The cluster set is always compact, and connected if D is locally connected at infinity. A sufficient condition for the latter is that any two points a, b ∈ D may be joined by a curve in D ∩{ z: | z | ≥ min{ | a |, | b | }}. For example, this is true for sectors, and also for sectors with mutually disjoint ‘holes’ | z − z j | ≤ r j , z j → ∞.

  2. 2.

    maple code decodedt1:=1+zˆ2-tˆ2; r1:=tˆ2+zˆ2; r2:=t+z; r:=r1/r2; t′ = 1 + z 2 −t 2; \(w = \mathfrak{r}(z,\mathsf{t}) = \frac{\mathfrak{r}_{1}(z,\mathsf{t})} {\mathfrak{r}_{2}(z,\mathsf{t})} = \frac{z^{2}+\mathsf{t}^{2}} {z+\mathsf{t}}\) s:=diff(r,z)+diff(r,t)∗t1; s1:=numer(s); s2:=denom(s);\(w' = \mathfrak{s}(z,\mathsf{t}) = \frac{\mathfrak{s}_{1}(z,\mathsf{t})} {\mathfrak{s}_{2}(z,\mathsf{t})}\) P:=resultant(x∗r2-r1,y∗s2-s1,t); P(z, w, w′) = 0Delta:=discrim(P,y); solve(Delta,x); compute the discriminant and its roots

  3. 3.

    The authors seemed to be unaware of Malmquist’s Second Theorem, or were in doubt about his reasoning. One reason might be that several elegant and transparent proofs of his First Theorem were known, but none for his Second. Eremenko’s commented in [35]: “The paper [42] (Malmquist’s paper [114] here) has had practically no influence on the work of other authors […]”.

  4. 4.

    This is a most non-trivial fact, known as Tsen’s Theorem (C. Tsen, Divisionsalgebren über Funktionenkörpern, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1933), 335–339), communicated to me by A. Eremenko. A local form generalising Tsen’s Theorem was proved by Lang (On quasi algebraic closure, Ann. of Math. 55 (1952), 373–390).

  5. 5.

    Equations (5.22) and (5.23) correspond to w = R(e az) when (5.19) has genus zero.

  6. 6.

    One cannot expect to re-construct the parametrisation \(w = \mathfrak{r}(z,\mathsf{t})\) and \(\mathsf{t}' =\hat{ P}(z) -\mathsf{t}^{2}\). The fact that \(\mathsf{t},w \in \mathfrak{Y}_{1,1}\) and \(\mathfrak{w} = \frac{\coth ^{2}\mathfrak{z}} {\coth \mathfrak{z}-2}\) suggests \(w = \frac{\mathsf{t}^{2}} {\mathsf{t}-2z-c_{0}}\) and \(\hat{P}(z) = z^{2} + a_{1}z + a_{0}\), actually a 1 = a 0 = c 0 = 0.

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Steinmetz, N. (2017). Algebraic Differential Equations. In: Nevanlinna Theory, Normal Families, and Algebraic Differential Equations. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-59800-0_5

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