A Computational Approach to Polynomial Conservation Laws

Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu*, Hamid Rahkooy, Matthias Sei, Thomas Sturm

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies.

Original languageEnglish
JournalSIAM Journal on Applied Dynamical Systems
Issue number1
Pages (from-to)813-854
Number of pages42
Publication statusPublished - 2024

Bibliographical note

Publisher Copyright:
© 2024 Society for Industrial and Applied Mathematics.


  • chemical reaction networks
  • comprehensive Gröbner systems
  • first integrals
  • polynomial conservation laws
  • polynomial ODEs
  • syzygies


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