Algorithms for Discrete Differential Equations of Order 1

Discrete differential equations of order 1 relate polynomially a power series F(t,u) in t with polynomial coefficients in a ''catalytic'' variable~u and one of its specializations, say F(t,u). Such equations are ubiquitous in combinatorics, notably in the enumeration of maps and walks. When the solution F is unique, a celebrated result by Bousquet-Mélou and Jehanne, reminiscent of Popescu's theorem in commutative algebra, states that F is algebraic. We address algorithmic and complexity questions related to this result. In generic situations, we first revisit and analyze known algorithms, based either on polynomial elimination or on the guess-and-prove paradigm. We then design two new algorithms: the first has a geometric flavor, the second blends elimination and guess-and-prove. In the general case (no genericity assumptions), we prove that the total arithmetic size of the algebraic equations for $F(t,1)$ is bounded polynomially in the size of the input discrete differential equation, and that one can compute such equations in polynomial time.