Quadratization of ODEs: Monomial vs. Non-Monomial

Overview

Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. It has been recently used as a preprocessing step for new model order reduction methods, so it is important to keep the number of new variables small. Several algorithms have been designed to search for a quadratization with the new variables being monomials in the original variables. To understand the limitations and potential ways of improving such algorithms, we study the following question: can quadratizations with not necessarily new monomial variables produce a model of substantially smaller dimension than quadratization with only new monomial variables? To do this, we restrict our attention to scalar polynomial ODEs. In our first result, we provide a necessary and sufficient condition on when one new variable is enough to quadratize. Secondly, we show that all degree 6 scalar polynomial ODEs can be quadratized with exactly two new non-monomial variables, an improvement on three new monomial variables. Based on these results, we observe that a quadratization with not necessarily new monomial variables can be much smaller than a monomial quadratization even for scalar ODEs. The main results of the paper have been discovered using computational methods of applied nonlinear algebra (Gröbner bases), and we describe these computations.

Competition history

• ISEF 2021 Mathematics · Entry MATH035

Resources

• ISEF abstract page ↗

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