Quadratization of ODEs: Monomial vs. Non-Monomial

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 pre-processing 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 monomial new variables produce a model of substantially smaller dimension than quadratization with only monomial new variables? To do this, we restrict our attention to scalar polynomial ODEs. Our first result is that a scalar polynomial ODE $\dot{x}=p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots + a_0$ with $n\geqslant 5$ and $a_n\neq0$ can be quadratized using exactly one new variable if and only if $p(x-\frac{a_{n-1}}{n\cdot a_n})=a_nx^n+ax^2+bx$ for some $a, b \in \mathbb{C}$. In fact, the new variable can be taken $z:=(x-\frac{a_{n-1}}{n\cdot a_n})^{n-1}$. Our second result is that two non-monomial new variables are enough to quadratize all degree $6$ scalar polynomial ODEs. Based on these results, we observe that a quadratization with not necessarily monomial new 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 (Gr\"obner bases), and we describe these computations.