Treffer: Best linear common divisors for approximate degree reduction

Let \(f_ 1,\dots,f_ m\) be a set of polynomials of highest degree \(n\). One can search for perturbation polynomials \(\varepsilon_ j\) of degree \(\leq n\) such that \(f_ 1 + \varepsilon_ 1,\dots,f_ m + \varepsilon_ m\) have a common zero. The size of the \(m\)-vector \(\vec\varepsilon(t)\) with components \(\varepsilon_ j(t)\) is measured by the norm \(\| \vec\varepsilon\| := \max\{|\vec\varepsilon(t)|, t \in [a,b]\}\), where \(|\vec\varepsilon(t)|\) is the Euclidean norm and \([a,b]\) is a fixed interval. It is shown that this problem with the side condition that the size \(\| \vec\varepsilon\|\) of the perturbation be minimal can be solved analytically. This is applied to the problem of approximately reducing the degree of a rational curve.