Stratified order one differential equations in positive characteristic
This paper continues previous studies on the classification of first-order differential equations over \(C\left( z \right)\) and finding algebraic solutions of such equations see [\textit{Ngo, L.X. Chau, K. A. Nguyen} and \textit{M. van der Put}, ``Equivalence of differential equations of order one'', Preprint, Top, J., \url{arXiv:1303.4960}, (2013)]. Let \(C\) be an algebraically closed field of characteristic \(p>0\), and \(K\) be a finite separable extension of \(C\left( z \right)\). The authors consider equations of the form \(f\left( {y}',y \right)=0\), where \(f\in K\left[ S,T \right],\text{ }\,f\) is absolutely irreducible polynomial and \(\frac{df}{dS}\notin (f)\). In addition the natural continuation of the differentiation of \(K\) onto \(K\left[ S,\text{ }T \right]/(f)\) must satisfy the condition \({{D}^{p}}=0\) (admits a stratification). They give criteria for deciding whether the equation admits a stratification. It turns out that the autonomous (\(f\in C\left[ S,\text{ }T \right]\)) differential equation admitting a stratification has a separable algebraic solution. Also for such equations a Grothendieck-Katz conjecture is true, see \textit{M. van der Put} [Indag. Math., New Ser. 12, No. 1, 113--124 (2001; Zbl 1004.12005)]. All solutions of \(f\left( {y}',y \right)=0\) are algebraic if and only if for almost all primes \(p\) the reduced equation \(f \mod p\) is stratified.
scientific article; zbMATH DE number 6378453
scientific article