AI 中文总结
本文针对变量分离的有理函数系数向量场，证明其代数可积性的充要条件，验证相关广义猜想，最终给出这类向量场及其一积分的系数显式公式。
AI 中文摘要
我们在特征为零的域上开展研究，主要是在有理数域的代数闭包上进行。对于系数为变量分离的有理函数的向量场，我们证明了其代数可积性的充要条件，即被该向量场零化的有理函数子环（其一积分环）具有最大可能的维数。我们通过对几乎所有素数模下的向量场进行算术约化来完成这一证明。特别地，我们验证了由这类向量场定义的叶状结构的广义Grothendieck-Katz p-曲率猜想。最后，我们利用该验证结果给出所有变量分离的代数可积向量场及其一积分的系数显式公式。
英文摘要
We work over a field of characteristic zero - primarily over an algebraic closure of the field of rational numbers. We prove a necessary and sufficient condition for a vector field whose coefficients are rational functions separating variables to be algebraically integrable, that is, the subring of rational functions killed by this vector field, its ring of first integrals, is of maximal possible dimension. We do it arithmetically by reducing the vector field modulo almost all primes. In particular, we verify the generalized Grothendieck--Katz p-curvature conjecture for foliations defined by these vector fields. Finally, we use the outcome of that verification to provide explicit formulas for coefficients of all algebraically integrable vector fields separating variables, and their first integrals.
Comments18 pages; Comments welcome!
