From linear recurrence relations to linear ODEs with constant coefficients (Q2803561)

Linear Ordinary Differential Equations (ODEs) with constant coefficients are studied by looking in general at linear recurrence relations in a module with coefficients in an arbitrary \(\mathbb{Z}\)-algebra. The bridge relating the two theories is the notion of \textit{formal Laplace transform} associated to a sequence of invertible elements of the base ring. From this more economical perspective, \textit{generalized Wronskians} associated to solutions of linear ODEs are revisited, mentioning their relationships with \textit{Schubert Calculus} for Grassmannians.NEWLINENEWLINELet \(r\) be a positive integer and let \(B_r=\mathbb Z[e_1,\dots, r_r]\). Let \(A\) be a commutative associative \(B_r\)-algebra with a unit. Let \(M\) be an \(A\)-module. \(\mathrm{Hom}_A(A[X], M)\) has a natural structure of an \(A[X]\)-module, with the multiplication defined by the rule \((f\cdot v)(g)=v(fg)\), \(f, g\in A[X]\), \(v\in \mathrm{Hom}_A(A[X], M)\).NEWLINENEWLINEFor fixed \(f\in A[X]\) and \(w\in \mathrm{Hom}_A(A[X], M)\), the authors study the equation \(f\cdot y=w\) and solve it (Theorem~4.5) for arbitrary initial condition \((X^iy)(1)=y(X^i)=\psi(X^i)\), \(0\leq i\leq \deg(f)-1\), \(\psi\in \mathrm{Hom}_A(A[X], M)\). More precisely, for a fixed degree \(r\in \mathbb Z\), \(r\geq 1\), the authors solve the universal equation \(p_r(X)\cdot y=w\), \(p_r(X)=X^r-e_1X^{r-1}+\dots+(-1)^re_r\).NEWLINENEWLINEFix a sequence \(\pmb \alpha=(\alpha_0,\alpha_1, \dots)\) of invertible elements of \(A\), fix an indeterminate \(D_{\mathbf \alpha}\), introduce on \(\mathrm{Hom}_A(A[X], M)\) a structure of an \(A[D_{\pmb \alpha}]\)-module by NEWLINE\[NEWLINE (D_{\pmb \alpha}\cdot v)(X^n)=\alpha_{n+1}\alpha_n^{-1}v(X^{n+1}), NEWLINE\]NEWLINE and denote this module by \(\mathrm{Hom}_A(A[X], M)_{\pmb \alpha}\).NEWLINENEWLINEThe \(\pmb \alpha\)-formal Laplace transform establishes an isomorphism, i.~e., \(L_{\pmb \alpha} D_{\pmb \alpha}=XL_{\pmb \alpha}\), NEWLINE\[NEWLINE L_{\pmb \alpha}: \mathrm{Hom}_A(A[X], M)_{\pmb \alpha}\to \mathrm{Hom}_A(A[X], M),\quad L_{\pmb \alpha}(v)(X^n)=\alpha_n v(X^n). NEWLINE\]NEWLINE This establishes a link to the linear ODEs as for \(\pmb \alpha=(0!, 1!, 2!,\dots)\) the multiplication by \(D_{\pmb \alpha}\) is just the standard derivation if one identifies \(\mathrm{Hom}_A(A[X], M)\) with \(M[[t]]\).NEWLINENEWLINEThe authors introduce generalized Wronskian determinants, which are elements in \(A[X]^\vee=\mathrm{Hom}_A(A[X], A)\), and prove that they satisfy residue formulas very similar to the Giambelli's equalities from the Schubert calculus.NEWLINENEWLINEIn the case when \(A\) is the Chow ring of the complex Grassmanian \(G(r, n)\) of \(r\)-dimensional subspaces in \(\mathbb C^n\), this establishes an \(A\)-module isomorphism between \(A\) and the module generated by the generalized Wronskians, each generalized Wronskian corresponding to a Schubert cycle.NEWLINENEWLINEThe paper under review consists of six sections. Section~1 is an introduction, the main result of the paper is formulated here. In section~2 some preliminaries and notations are introduced. In section~3 the authors study some isomorphisms of modules that are used in the paper. The kernel of the multiplication operator \(\mathrm{Hom}_A(A[X], M)\rightarrow{p_r(X)\cdot } \mathrm{Hom}_A(A[X], M)\) is described here. Section~4 deals with the algebraic Cauchy problem mentioned above, Theorem~4.5 presents a universal formula for its solution. Some examples are considered here. Formal Laplace transform is discussed in section~5. The solution of the Cauchy problem is reformulated in terms of the Laplace transform. The authors revisit the examples from the previous section in the case \(\pmb\alpha =(0!, 1!, 2!,\dots)\), i.~e., for linear ODEs. Section~6 discusses generalized Wronskian determinants and their algebraic treatment. A purely algebraic version of the theorem by Abel and Liouville is mentioned here. The links to the Boson-Fermion correspondence and to the Schubert calculus are discussed as well.