Solutions of algebraic differential equations are conjectured to be algebraic exactly when their Taylor coefficients have almost no primes in their denominators, and the conjecture is proved for Picard-Fuchs equations and for isomonodromy equations such as Painlevé VI.
Algebraic solutions of linear differential equations: an arithmetic approach
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.
fields
math.AG 1years
2025 1verdicts
CONDITIONAL 1representative citing papers
citing papers explorer
-
Algebraicity and integrality of solutions to differential equations
Solutions of algebraic differential equations are conjectured to be algebraic exactly when their Taylor coefficients have almost no primes in their denominators, and the conjecture is proved for Picard-Fuchs equations and for isomonodromy equations such as Painlevé VI.