Foundations of Computational Mathematics

, Volume 18, Issue 2, pp 291–308

# The Differential Counting Polynomial

• Markus Lange-Hegermann
Article

## Abstract

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common generalization of the dimension polynomial and the (algebraic) counting polynomial. Under mild additional assumptions, the differential counting polynomial decides whether a given set of solutions of a system of differential equations is the complete set of solutions.

## Keywords

Differential algebra Dimension polynomial Differential counting polynomial Thomas decomposition

## Mathematics Subject Classification

12H05 35A01 35A10 34G20

## Notes

### Acknowledgements

The author was partly supported by the DFG Schwerpunkt SPP 1489 and Graduiertenkolleg Experimentelle und konstruktive Algebra of the DFG. His gratitude goes to the anonymous referees for a thorough review with many valuable comments.

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