论文标题
odes的二次化：单一与非公主
Quadratization of ODEs: Monomial vs. Non-Monomial
论文作者
论文摘要
二次化是通过引入新变量的多项式右侧的ODES系统的变换，最多是二次右侧的ODES系统。它最近被用作新模型订购方法的预处理步骤，因此将新变量的数量保持较小非常重要。已经设计了几种算法来搜索二次化，而新变量是原始变量中的单一变量。为了了解改善此类算法的局限性和潜在方法，我们研究以下问题：不一定是单一新变量的二次化可以产生一个比仅具有单一新变量的二次化尺寸要小得多的模型？ 为此，我们将注意力限制在标量多项式ODE上。我们的第一个结果是标量多项式$ \ dot {x} = p（x）= a_nx^n+a_+a_ {n-1} x^{n-1}+\ ldots+a_0 $ at $ n \ geqslant 5 $ and $ a_n \ neq0 $ a_n \ neq0 $可以使用一个新的可变。 $ p（x- \ frac {a_ {n-1}} {n \ cdot a_n}）= a_nx^n+ax^2+ax^2+bx+bx+bx+for Some $ a，b \ in \ Mathbb {c} $。实际上，可以采用新变量$ z：=（x- \ frac {a_ {n-1}}} {n \ cdot a_n}）^{n-1} $。我们的第二个结果是，两个非公主的新变量足以使所有程度$ 6 $标量的多项式ODE相称。基于这些结果，我们观察到，即使对于标量odes，具有单一新变量的二次化不一定要比单一二次化小得多。 本文的主要结果是使用应用的非线性代数（GröbnerBases）的计算方法发现的，我们描述了这些计算。
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 (Gröbner bases), and we describe these computations.
