Spectral radius of a derivation and algebraic extensions
Let \(F\) be a complete ultrametric valued field of unequal characteristic and divisble valuation group, and let \(D\) be a continuous derivation of \(F\). We assume that the valuation \(|\cdot|\) of \(F\) is not trivial on the field of constants \(C\) of \(D\). The operator norm \(||D||\) is the supremum over \(a \neq 0\) of the ratios \(|D^n(a)|/|a|\), and the spectral radius of \(D\) is the infimum of the sequence \(||D^n||^{\frac{1}{n}}\). We denote the spectral radius of \(D\) by \(R_F\). Let \(K \supseteq F\) be a finite field extension. The valuation of \(F\) and the derivation \(D\) are extended to \(K\). The author is interested in the relation between the spectral radii \(R_K\) and \(R_F\), and obtains the following results: NEWLINENEWLINENEWLINEIf the residue field extension \(\overline{K} \supseteq \overline{F}\) is separable, then \(R_K=R_F\). NEWLINENEWLINENEWLINEIf the residue field extension is inseparable of degree \(p\) equal to the characteristic of \(\overline{F}\), then \(R_K \geq |p|^{\frac{p}{p-1}}R_F\). NEWLINENEWLINENEWLINEThe proof uses the fact that the derivation \(D\) provides a homomorphism (``Taylor Embedding'') from \(F\) to the formal power series \(FX\); the author identifies the image and puts a valuation on it that makes the Taylor embedding an isometry.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].
scientific article; zbMATH DE number 1665080