Taylor approximations of multidimensional linear differential systems

ABSTRACT

The Taylor approximations of a multidimensional linear differential system are of importance as they contain a complete information about it. It is shown that in order to construct them it is sufficient to truncate the exponential trajectories only. A computation of the Taylor approximations is provided using purely algebraic means, without requiring explicit knowledge of the trajectories.

KEYWORDS:

• Multidimensional system
• Taylor approximation
• polynomial-exponential trajectory
• proper kernel representation

Disclosure statement

No potential conflict of interest was reported by the author.

Appendix. Description via cohomologies

In this appendix, we want to give a cohomological description of Taylor approximations. (It will be assumed that the reader has some familiarity with coherent sheaves and their cohomologies.) We keep the notation of the previous sections.

Let s₀ be a ‘homogenising’ indeterminate, and let $\mathbb{C}[s_0,s]$ denote the ring of homogeneous polynomials (in s₀, s₁,… , s_n). The latter defines an n-dimensional projective space, the structure sheaf of which denote by $\mathcal{O}$.

For each k ≥ 0, denote by $\mathbb{C}{[s_0,s]}_k$ the space of homogeneous polynomials of degree k. For a polynomial $g\in \mathbb{C}{\left[s\right]}_{\le k}$, the polynomial $s_0^{k}g(s/s_0)\in \mathbb{C}{[s_0,s]}_k$ is called the k-homogenisation of g.

The homogeneous matrix $R^h=\text{diag}(s_0^{a_1},...,s_0^{a_p})R(s/s_0)$ determines a sheaf homomorphism $$R^h:{\mathcal{O}}^q\to \underset{i=1}{\overset{p}{⨁}}\mathcal{O}\left(a_i\right).$$

If k ≥ 0, then $H^n\mathcal{O}(-k-n-1)$ is a linear space with basis $$\left\{s_0^{-i_0-1}\cdots s_n^{-i_n-1}|i_{\alpha }\ge 0,\sum i_{\alpha }=k\right\}$$(see Section III.5 of Hartshorne's book Hartshorne (1977)). For every k ≥ 0, we therefore have a canonical isomorphism $$\mathbb{C}{\left[t\right]}_{\le k}\simeq H^n\mathcal{O}(-k-n-1),$$given by $$t_1^{i_1}...t_n^{i_n}\mapsto s_0^{(i_1+\cdots +i_n-k-1)}{s}_1^{-i_1-1}...s_n^{-i_n-1}.$$

Lemma 3:

Let r be a polynomial and l a nonnegative integer such that $deg\left(r\right)\le l$, and let r^h = s^l₀r(s/s₀) be the l-homogenisation of r. Then, the diagram $$\begin{array}{ccc}\mathbb{C}{\left[t\right]}_{\le k}& \to & \mathbb{C}{\left[t\right]}_{\le k-l}\\ \downarrow & & \downarrow \\ H^n\mathcal{O}(-k-n-1)& \to & H^n\mathcal{O}(l-k-n-1)\end{array}$$is commutative. (Here, the top arrow is τ_{k − l} ○ Π ○ r, and the bottom one is induced by r^h.)

Proof:

It suffices to consider the case when r = s^m with |m| ≤ l.

The space $\mathbb{C}{\left[t\right]}_{\le k}$ is generated by elements tⁱ with |i| ≤ k. Take any such element tⁱ. Then, $${\tau }_{k-l}\Pi \left(s^m{t}^i\right)=\left\{\begin{array}{cc}{t}^{i-m}& \mathrm{if}i\ge m\mathrm{and}|i-m|\le k-l;\hfill \\ 0& \mathrm{otherwise}.\hfill \end{array}\right.$$

We have r^h = s^{l − |m|}₀s^m. The bottom arrow above takes $s_0^{\left|i\right|-k-1}{s}_1^{-i_1-1}...s_n^{-i_n-1}$ to $$s_0^{l-k+\left|i\right|-\left|m\right|-1}{s}_1^{m_1-i_1-1}...s_n^{m_n-i_n-1}$$if i ≥ m and |i| − |m| ≤ k − l, and to 0 otherwise.

One easily completes the proof.

Theorem 3:

For every k ≥ 0, ${\mathcal{B}}_{|k}$ is canonically isomorphic to the kernel of the linear map $$H^n{\mathcal{O}}^q(-k-n-1)\to \underset{i=1}{\overset{p}{⨁}}{H}^n\mathcal{O}(a_i-k-n-1).$$

Proof:

By the previous lemma, the diagram $$\begin{array}{ccc}\mathbb{C}{\left[t\right]}_{\le k}^q& \to & \underset{i=1}{\overset{p}{⨁}}\mathbb{C}{\left[t\right]}_{\le k-a_i}\\ \downarrow & & \downarrow \\ H^n{\mathcal{O}}^q(-k-n-1)& \to & \underset{i=1}{\overset{p}{⨁}}{H}^n\mathcal{O}(a_i-k-n-1)\end{array}$$is commutative. (Here, the top arrow is ρ_k, and the bottom one is HⁿR^h( − k − n − 1).) The downward arrows are isomorphisms. So, the assertion follows from Theorem 2.

Remark 4:

The module associated with a linear differential system admits a natural sheafification so that the connection with Algebraic Geometry is not incidental. The linear maps of Theorem 3 have appeared for one-dimensional case in Lomadze (2001).