Autonomous functions (Q1121414)

A function y(x) is said to be differentially algebraic if it satisfies a nontrivial algebraic differential equation \(P(x,y)\equiv P(x,y(x),y'(x),...,y^{(n)}(x))=0\) where P is a polynomial in its \(n+1\) variables with complex coefficients. A function f(x) is said to be autonomous if it is differentially algebraic and every translate \(f_ s=f(t-s)\) satisfies every algebraic differential equation which is satisfied by f. The paper deals with the following: 1. Equivalent definitions of an autonomous function; 2. Sums and products of autonomous functions; 3. Composition with autonomous entire functions; 4. Finite sums of periodic functions; 5. Open problems; 6. A short and accessible proof of the Kochin-Ostrowski theorem.