Ritt-Wu characteristic set method for Laurent partial differential polynomial systems
Let \(F\) be a field with \(m\) commuting derivations \(\{\delta_1, \dots, \delta_m\}\). Let \(\mathbb Y=\{y_1, \dots, y_n\}\) be differential indeterminates and \(F\{\mathbb Y\}\) the resulting differential ring. Let \(\Theta\) be the set of monic monomials in the \(\delta_i\) and let \(\Theta(\mathbb Y)\) be the set of all \(\theta y_j\) for \(\theta \in \Theta\) and \(1 \leq j \leq n\). Assume that \(\Theta(\mathbb Y)\) has been given a total order such that for any \(\alpha\), \(\beta\) and any \(i\) \(\delta_i(\alpha) \alpha\) and if \(\alpha \beta\) then \(\delta_i(\alpha) \delta_i(\beta)\). For \(f \in F\{\mathbb Y\}\), the leader of \(f\), \(\text{ld}_f\), is the largest element of \(\Theta(\mathbb Y)\) occurring in \(f\). If also \(g \in F\{\mathbb Y\}\), no proper derivative of \(\text{ld}_g\) occurs in \(f\), and as polynomials in \(\text{ld}_f\) (respectively \(\text{ld}_g\)) \(f\) has smaller degree then \(f\) is reduced with respect to \(g\). If \(S\) is a subset of \(F\{\mathbb Y\}\) then \(f\) is reduced with respect to \(S\) is it is reduced with respect to every element of \(S\). \(S\) is autoreduced if each element is reduced with respect to the set of the rest of the elements of \(S\). A differential chain is an autoreduced (finite) set whose elements are strictly increasing in rank. A polynomial \(f \in F\{\mathbb Y\}\) is monomial-primitive if it is not a multiple of any element of \(\Theta(\mathbb Y)\). The Laurent differential polynomial ring \(F\{\mathbb Y^\pm\}\) is the polynomial ring in the set of all \(\theta y_j, (\theta y_j)^{-1}\), regarded as a differential ring with the \(\delta_i\) extended by the quotient rule. Any \(f \in F\{\mathbb Y^\pm\}\) has an associated monomial-primitive normal form. A Laurent differential chain is a differential chain in \(F\{\mathbb Y\}\) all of who members are monomial-primitive. Let \(\mathbb P\) be a subset of \(F\{\mathbb Y^\pm\}\). A characteristic set for \(\mathbb P\) is a Laurent differential chain which is a characteristic set for the subset of \(F\{\mathbb Y\}\) made up of the normal forms of the elements of \(\mathbb P\). The authors define the property of regular for Laurent differential chains, and prove that a Laurent differential chain is regular when it is a certain kind of characteristic set. They show that regularity can be reduced to the examination of each \(y_j\) separately. Then they further show that regularity in the single variable case can be reduced to looking at invertibility with respect to certain specific differential polynomials. For the order \(1\) case they provide a decision procedure.
Laurent partial differential polynomial system
scientific article; zbMATH DE number 7032428
scientific article