Quadratization of ODEs: Monomial vs. Non-Monomial
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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 ẋ=p(x)=a_nx^n+a_n-1x^n-1+… + a_0 with n⩾ 5 and a_n≠0 can be quadratized using exactly one new variable if and only if p(x-a_n-1/n· a_n)=a_nx^n+ax^2+bx for some a, b ∈ℂ. In fact, the new variable can be taken z:=(x-a_n-1/n· 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 (Gröbner bases), and we describe these computations.
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