Dimension polynomials of intermediate fields and Krull-type dimension of finitely generated differential field extensions
Let \(K\) be a (partial) differential field of characteristic \(0\) with set of derivations \(\Delta\), let \(L=K\langle\eta_1, \dots, \eta_s\rangle\) be a finitely generated differential field extension of \(K\) and let \(F\) be an intermediate differential field. Generalizing Kolchin's differential dimension polynomial, the author shows that there is a rational polynomial integral on the integers which counts the transcendence degree of \(F_r\) over \(K\), where \(F_r\) is the intersection of \(F\) with the extension of \(K\) algebraically generated by derivatives of the \(\eta_i\) up to order \(r\). As in Kolchin's case, the polynomial depends on the \(\eta_i\)'s, but not its degree, \(d\), and leading coefficient \(b_d\). Let \(m=| \Delta |\). Then \(d \leq m\); it is convenient to write the polynomial including terms up to degree \(m\), in which case the coefficient \(b_m\) is also an invariant: either \(d < m\) and \(b_m=0\) or \(b_m=d\). In any event, \(b_m\) is the differential transcendence degree of \(F\) over \(K\). The theorem is proved via a consideration of Hilbert functions of differential modules. Retain the above notation for \(L\) and \(K\). The author further considers the set \(\mathfrak U\) of pairs \((F,E)\) of intermediate differential fields with \(F \supseteq E\). The author shows that there is a unique mapping \(\mu: \mathfrak U \to \mathbb Z \cup \{\infty\}\), which takes values \(-1\) or larger, and such that if \(k\) is any natural number then \(\mu(F,E) \geq k\) if and only if \(F\) is not algebraic over \(E\) and there exists an infinite proper descending chain \(\{ F_i\}\) of intermediate differential fields with \(F_0=F\), \(F_i \supseteq E\) all \(i\), and \(\mu(F_i,F_{i+1}) \geq k-1\) all \(i\). He then defines the differential transcendence type of \(L\) over \(K\), \(\Delta\text{-tr.type}(L/K)\), to be the supremum of \(\mu\). He also defines the differential transcendence dimension of \(L\) over \(K\), \(\Delta-tr.dim(L/K)\), to be the supremum of the \(q\)'s for which there is a chain of intermediate differential fields \(F_0 \supset \dots \supset F_q\) such that \(\mu(F_{i-1}, F_i)=\Delta\text{-tr.type}(L/K)\) for \(1 \leq i \leq q\). In (all) the above notations, the author proves that \(\Delta\text{-tr.type}(L/K) \leq d \leq m\); if the differential transcendence degree of \(L\) over \(K\) is positive then \(\Delta\text{-tr.type}(L/K)=m\) and \(\Delta\text{-tr.dim}(L/K)\) coincides with the differential transcendence degree; and if the differential transcendence degree is zero then \(\Delta\text{-tr.type}(L/K) < m\).
scientific article; zbMATH DE number 5905248