The Grothendieck conjecture and Padé approximations
The Grothendieck Conjecture (G. C.) asserts that a differential operator \(L\in K(X)[d/dX]\) having a full set of polynomial solutions modulo p, for almost all primes p of the number field K, has a full set of solutions in the algebraic closure of K(X). The authors have proved elsewhere that under the assumptions of the G. C. if f(X) is a power series solution of L at a regular point \(\zeta\in K\), and \(f^ i(X)=\sum^{\infty}_{n=0}a_{n,i} (X-\zeta)^ i\) then the size of \(a_{n,i}\) and the minimal common denominator of \(\{a_{0,i}\), \(a_{1,i},...\), \(a_{n,i}\}\), \(i=0,1,...,m\), are bounded by \(C^ n_ 0\) for \(C_ 0=C_ 0(f,\zeta)\) and \(n\geq n_ 0(m)\). The authors then consider Padé approximations to 1, f(x),..., \(f^{m-1}(x)\) at \(x=\zeta:\) they obtain polynomials \(P_ 0,...\), \(P_{m-1}\) such that \(R(x)=\sum^{m-1}_{i=0}P_ i f^ i\) has a high zero at \(x=\zeta\), while the coefficients of the \(P_ i's\) are bounded in size. The following assumption (satisfied in many geometrically important cases: first order equations over curves, Lamé equations,...) is now crucial: that X and f(X) can be parametrized in a neighborhood of \(X=\zeta\) by a pair of meromorphic functions \(X=g(u)\), \(f(X)=h(u)\) with \(g^{-1}(\zeta)'\neq \infty\). When this is true a contradiction is obtained to the product formula for the first non-zero coefficient of R(x) at \(x=\zeta\). Therefore \(R(X)=0\) and f is algebraic over K(X). So G. C. is proved in many new cases.
scientific article; zbMATH DE number 3916417