r(x,y) + q(x,y) p(x,y) = f(x) - g(y)

by Dr Manfred Buchacher (Johannes Kepler Universität Linz)

Thursday Jun 12, 2025, 4:00 PM → 6:00 PM Europe/Paris

207 (Bat 1R2)

I will talk about my research on finding the solutions of such equations when r(x,y), q(x,y), p(x,y), f(x) and g(y) are rational functions of which the (irreducible) polynomial p(x,y) and the rational function r(x,y) are known.

 

At first glance, it might not be clear what the meaning of the equation is and why one should find it interesting. However, there is a field theoretic interpretation that is helpful. Let C(x,y) be the field over C that is generated by elements x and y satisfying the relation p(x,y) = 0, and let C(x) and C(y) be the subfields generated by x and y, respectively. Then the above equation has a solution if and only if the element r(x,y) of C(x,y) is an element of C(x) + C(y). Note that when r = 0, the problem is the problem of computing the intersection of C(x) and C(y), and that when g = 0, finding a solution of the equation answers whether r(x,y) is an element of C(x) and how it can be expressed in terms of x.

 

The problem originally arose in enumerative combinatorics, in the enumeration of restricted lattice walks. Other applications are parameter identification problems in ODE models, or problems of image recognition, for instance. There are most likely many more as it is a basic problem in computational algebra and algebraic geometry. 

 

I will present a semi-algorithm that solves such equations. "Semi" means that it might not terminate. It terminates if the equation has a non-trivial solution. However, if there is no such solution, it might not. Its termination depends on a dynamical system related to the curve associated with p(x,y) and the location of the poles of r(x,y).

 

The methods and arguments employed are mostly elementary, and come from linear algebra, the Newton-Puiseux algorithm, and Galois theory, and can be interpreted as a combination of tropical (in the sense of tropical geometry) and Galois theoretic arguments. The question how the semi-algorithm could be turned into an algorithm connects it to number theory and problems in arithmetic / algebraic dynamics.

 

There will be pictures.