On the geometric degree of the prolongation varieties of the differential system x₁'=x₂², x₂'=x₁²

@misc{puddu2026geometric,
  title = {{On the geometric degree of the prolongation varieties of the differential system $x_1'=x_2^2$, $x_2'=x_1^2$}},
  author = {Mattia Puddu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06638},
  eprint = {2611.06638},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06638},
}

TY  - UNPB
TI  - On the geometric degree of the prolongation varieties of the differential system x_1'=x_2^2, x_2'=x_1^2
AU  - Mattia Puddu
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06638
UR  - https://arxiv.org/abs/2611.06638
AB  - 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.
ER  -