On the geometric degree of the prolongation varieties of the differential system $x_1'=x_2^2$, $x_2'=x_1^2$

Abstract

A prolongation variety of a polynomial differential system $S$ is an algebraic variety defined by the equations of $S$ and their derivatives up to a given order. We compute the geometric degree of the prolongation varieties associated with the system $x_1'=x_2^2$, $x_2'=x_1^2$ in terms of the prolongation order, thereby proving a conjecture stated by Gleb Pogudin in Section 2, Question 6 of "Some Open Problems in Differential Algebra". Our approach is motivated from the beginning by doing computational experiments in OSCAR in order to find the number of solutions of a zero-dimensional system using algebraic Thomas decomposition. The result will follow from a coprimality condition on a family of polynomials with integer coefficients, which is verified by showing that their roots satisfy a strong interlacing property. Finally, we propose an extension of the conjecture for the systems $x_1'=x_2^n$, $x_2'=x_1^n$ with $n>2$.