Quadratization of ODEs: Monomial vs. Non-Monomial

Quadratization is a transform of system ODEs with polynomial right-hand side into at most quadratic via the introduction new variables. It has been recently used as pre-processing step for model order reduction methods, so it important to keep number variables small. Several algorithms have designed search quadratization being monomials in original To understand limitations and potential ways improving such algorithms, we study following question: can quadratizations not necessarily monomial produce substantially smaller dimension than only variables? To do this, restrict our attention scalar ODEs. Our first result that ODE $\dot{x}=p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots + a_0$ $n\geqslant 5$ $a_n\neq0$ be quadratized using exactly one variable if $p(x-\frac{a_{n-1}}{n\cdot a_n})=a_nx^n+ax^2+bx$ some $a, b \in \mathbb{C}$. In fact, taken $z:=(x-\frac{a_{n-1}}{n\cdot a_n})^{n-1}$. second two non-monomial are enough quadratize all degree $6$ Based on these results, observe much even The main results paper discovered computational methods applied nonlinear algebra (Grobner bases), describe computations.