On differentially algebraic generating series for walks in the quarter plane

Abstract

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell–Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series.

Appendix A: Poles and residues

In this section, we collect various technical facts concerning the poles and residues of rational functions on $\overline{E_t}$, that is, elements of ${{\mathbb {C}}}(\overline{E_t})$. We will assume throughout this section that $\overline{E_t}$ is an elliptic curve endowed with two involutions ${\iota _1},{\iota _2}$. We denote by P the point of $\overline{E_t}$ such that $\tau ={\iota _2}\circ {\iota _1}$ is the translation by P. In our discussions below, we need to expand elements of ${{\mathbb {C}}}(\overline{E_t})$ in power series at points of $\overline{E_t}$ and compare the expansions at various points. In order to do this in a consistent way the following was introduced in [15]

Definition A.1

Let ${\mathcal {S}} = \{ u_Q \ | \ Q\in \overline{E_t}\}$ be a set of local parameters at the points of $\overline{E_t}$. We say S is a coherent set of local parameters if for any $Q \in \overline{E_t}$,

$$\begin{aligned} u_{\tau ^{-1}(Q)} = \tau (u_Q). \end{aligned}$$

Note that $\tau ^{-1}(Q) = Q\ominus P$, where $\ominus$ is subtraction in the group structure of the elliptic curve.

A coherent set of local parameters always exits. To see this, Let O be the origin of the group law on the elliptic curve $\overline{E_t}$ and, for any $Q \in \overline{E_t}$ let $\tau _Q$ be the translation by Q. The map $\tau _Q$ induces and isomorphism $\tau _Q: {{\mathbb {C}}}(\overline{E_t}) \rightarrow {{\mathbb {C}}}(\overline{E_t})$ (here we abuse notation and use the same symbol). Let t be a local parameter at O. The set of local paramters $\{ u_Q = \tau _{-Q}(t) \ | \ Q\in \overline{E_t}\}$ is a coherent set of local parameters.

Definition A.2

Let $u_Q$ be a local parameter at a point $Q \in \overline{E_t}$ and let $v_Q$ be the valuation corresponding to the valuation ring at Q. If $f \in { \mathbb {C}}(\overline{E_t})$ has a pole at Q or order n, we may write

$$\begin{aligned} f = \frac{c_{Q,n}}{u_Q^n} + \ldots + \frac{c_{Q,2}}{u_Q^2} + \frac{c_{Q,1}}{u_Q} + \tilde{f} \end{aligned}$$

where $v_Q(\tilde{f}) \ge 0$. We shall refer to $c_{Q,i}$ as the residue of order i at Q.

In the usual presentation of Riemann surfaces, one speaks of residues of meromorphic differential forms. These do not depend on the local parameters whereas any discussion of a powerseries expansion of a function at a point does depend on the local parameter. Fixing a set of local parameters allows the notion of residue of order i to be well defined.

The following definition is similar to Definition 2.3 of [12].

Definition A.3

Let $f \in k(\overline{E_t})$ and $S =\{ u_Q \ | \ Q\in \overline{E_t}\}$ be a coherent set of local parameters and $Q\in \overline{E_t}$. For each $j \in {{\mathbb {N}}}_{>0}$ we define the orbit residue of order j at Q to be

$$\begin{aligned} \mathrm{ores}_{Q,j}(f) = \sum _{i \in {{\mathbb {Z}}}} c_{Q\oplus iP, j.} \end{aligned}$$

Note that if $Q' = Q\oplus P$, then $\mathrm{ores}_{Q',j}(f) = \mathrm{ores}_{Q,j}(f)$ for any $j \in {{\mathbb {N}}}_{>0}$. Furthermore $\mathrm{ores}_{Q,j}(f) = \mathrm{ores}_{Q,j}(\tau (f))$. The following refines Proposition B.8 in [15] and is the reason for defining the orbit residue.

Proposition A.4

Let $b \in k(\overline{E_t})$ and $S =\{ u_Q \ | \ Q\in \overline{E_t}\}$ be a coherent set of local parameters. The following are equivalent.

• (1) There exists $g\in k(\overline{E_t})$ such that

  $$\begin{aligned} b = \tau (g) - g. \end{aligned}$$

• (2) For any $Q \in \overline{E_t}$ and $j \in {{\mathbb {N}}}_{>0}$

  $$\begin{aligned} \mathrm{ores}_{Q,j}(b) = 0. \end{aligned}$$

Proof

Proposition B.8 in [15] implies that (2) is equivalent to: there exists $Q\in \overline{E_t}$, $h \in {{\mathcal {L}}}(Q + \tau (Q))$ and $g \in \overline{E_t}$. such that $b = \tau (g) - g + h{.}$ Lemma 3.7 implies that this latter condition is equivalent to (1). $\square$

When applying Proposition A.4, we would like to verify the second condition using the fact that on a compact Riemann surface one has that the sum of the residues of a differential form is zero. Denoting by $\mathrm{Res}_Q\omega$ the usual residue at a point Q of a differential form $\omega$, we want to compare $\mathrm{Res}_Q(f\omega )$ with $c_{Q,1}$ where f is as in Definition A.2. To do this, we need to make a more careful selection of a coherent family of local parameters. For this, we will use the following lemma whose proof is similar to [10, Theorem 14, p. 127].

Lemma A.5

Let C be a nonsingular curve and $K = {{\mathbb {C}}}(C)$ its function field. Given a point $Q \in C$, a differential form $\omega$ regular and nonzero at Q, and integer $n \in {{\mathbb {N}}}$, there exists a local parameter $t_n \in K$ at Q such $\omega = (1+f)dt_n$ where $v_Q(f)>n$.

Proof

Let $t \in K$ be any local parameter at Q and let

$$\begin{aligned} \omega = (a_0 + a_1t + \ldots +a_nt^n + f_n) dt. \end{aligned}$$

where $f_n \in K$ and $v_Q(f_n) > n$. Let

$$\begin{aligned} t_n = a_0t + \frac{a_1}{2} t^2 + \ldots + \frac{a_n}{n+1} t^{n+1}. \end{aligned}$$

We then have that

$$\begin{aligned} \omega - dt_n = (a_0 + a_1t + \ldots +a_nt^n + f_n - \frac{dt_n}{dt}{)} dt =f_n dt. \end{aligned}$$

$\square$

Let $\Omega$ be a fixed regular differential form on $\overline{E_t}$. The maps ${\iota _1}, {\iota _2}, \tau = {\iota _2}{\iota _1}$ induce maps ${\iota _1}^*, {\iota _2}^*, \tau ^*$ on the space of differential forms. From [19, Lemma 2.5.1 and Proposition 2.5.2], we have that $\iota ^*_i(\Omega ) = -\Omega$ for $i=1,2$ and $\tau ^*(\Omega ) = \Omega$.

Definition A.6

Let $n \in {{\mathbb {N}}}$. We say that a coherent set $\{u_Q \ | \ Q\in \overline{E_t}\}$ of local parameters is n-coherent if for each $Q \in \overline{E_t}$, $\Omega = (1 + f_Q)du_Q$ where $v_Q(f_Q) > n$.

There always exists an n-coherent set of local parameters. To see this one modifies the construction following Definition A.1 by starting with a local parameter $t_n$ at O satisfying the conclusion of Lemma A.5 with respect to $\Omega$, that is, the order of $\Omega - dt_n$ at O is greater than n.

Fixed Assumption: Through the paper, we assume that when the kernel curve $\overline{E_t}$ is of genus one, we fix a 3-coherent set of local parameters ${\{u_{Q} \ | \ Q \in \overline{E_t}\}}$. The various elements that we consider will have poles of order at most 3 so we can always apply Lemmas A.7and A.9.

Having an n-coherent set of local parameters allows one to use the usual Residue Theorem.

Lemma A.7

Let $b\in {{\mathbb {C}}}(\overline{E_t})$ and assume that b has poles of order at most n at any point of $\overline{E_t}$. If $\{u_Q\}$ is an n-coherent set of local parameters, then for each $Q \in \overline{E_t}$, $\mathrm{Res}_{Q}(b\Omega )=c_{Q,1}$. Therefore, $\sum _{Q \in \overline{E_t}} c_{Q,1}=0$.

Proof

Since $\Omega = (1 + f_Q) du_Q$ with $v_Q(f_Q) > n$ we have

$$\begin{aligned} b\Omega = (\frac{c_{Q,n}}{u_Q^n} + \ldots + \frac{c_{Q,2}}{u_Q^2} + \frac{c_{Q,1}}{u_Q} + \tilde{f_Q}) du_Q \end{aligned}$$

where $v_Q(\tilde{f_Q}) > 0$. One now applies the usual Residue Theorem.$\square$

Remark A.8

• 1. In [16], the authors introduced the notion of a coherent set of analytic local parameters and showed that such a set exists on the universal cover of $\overline{E_t}$ and using these to induce such a set on $\overline{E_t}$. Alternatively, one can always find a coherent set of local parameters $\{u_Q \ | \ Q\in \overline{E_t}\}$ such that for each Q, $\Omega = du_Q$. One does this in the following way. If t is an analytic local parameter at O, we write $\Omega = \sum _{i = 0}^\infty a_i t^i dt$, $a_0 \ne 0$. The analytic function $u_0 = \sum _{i = 0}^\infty \frac{a_i}{i+1} t^{i+1}$ is an analytic local parameter at 0 and one can propagate this to become a coherent local family as above. Nonetheless, the $u_Q$ gotten in this way need not be in the function field of the curve since they are only defined locally. We introduce the notion of n-coherence to be able to stay in the algebraic setting.

• 2.  In [14], the authors uniformize the kernel curve E as a Tate curve, that is, as $C^*/q^\mathbb {Z}$ where C is an algebraically closed field extension of $\mathbb {Q}(t)$. In that setting, the field C(E) corresponds to the field ${\mathcal {M}} er (C^*)$ of meromorphic function over $C^*$ fixed by the automorphism $f(z) \mapsto f(qz)$ of ${\mathcal {M}} er (C^*)$. The first involution corresponds to $f(z) \mapsto f(1/z)$ and the automorphism $\tau$ to $f(z) \mapsto f(\tilde{q}z)$. The regular differential form on $C^*/q^\mathbb {Z}$ is $\frac{dz}{z}$ and the coherent set of local parameters given by the $u_{\overline{\alpha }}: \overline{z } \mapsto ln(\frac{z}{\alpha })$ for z close to $\alpha$ satisfies all the required properties.

The following summarizes useful properties of the $c_{Q,i}$ and the $\mathrm{ores}_{Q,j}(f)$ .

Lemma A.9

Let $n>1$ and $\{u_Q\}$ be an n-coherent set of local parameters. Assume $b \in {{\mathbb {C}}}(\overline{E_t})$ satisfy ${\iota _1}(b) = -b$.

• 1. For each $Q \in \overline{E_t}$, ${\iota _1}(u_Q) = -u_{{\iota _1}(Q)} +g_{{\iota _1}(Q)}$ where $v_{{\iota _1}(Q)}(g_{{\iota _1}(Q)}) > n+1$.

• 2. If

  $$\begin{aligned} b = \frac{c_{Q,n}}{u_Q^n} + \ldots + \frac{c_{Q,2}}{u_Q^2} + \frac{c_{Q,1}}{u_Q} + \tilde{f} \end{aligned}$$

  (A.1)

  where $v_Q(\tilde{f}) \ge 0$, then

  $$\begin{aligned} b =\frac{c_{\iota _{1}(Q),n}}{u_{\iota _{1}(Q)}^n} + \ldots + \frac{c_{\iota _{1}(Q),2}}{u_{\iota _{1}(Q)}^2} + \frac{c_{\iota _{1}(Q),1}}{u_{\iota _{1}(Q)}} + \tilde{g} \end{aligned}$$

  where $v_{\iota _{1}(Q)}(\tilde{g}) \ge 0$ and $c_{\iota _{1}(Q),j}=(-1)^{j+1}c_{Q,j}$ for any j. If follows that, if all the poles of b belong to the same $\tau$-orbit, then, for any even number j, we have $\mathrm{ores}_{Q,j}(b)=0$.

Proof

• 1. We have $\Omega = (1+f_Q) du_Q = (1+f_{{\iota _1}(Q)})d(u_{{\iota _1}(Q)})$ where $v_Q(f_Q)>n, v_{{\iota _1}(Q)}(f_{{\iota _1}(Q)}) >n$. Applying ${\iota _1}^*$ to the first equality we have

  $$\begin{aligned} -\Omega = {\iota _1}^*(\Omega ) = (1 + {\iota _1}(f_Q)){\iota _1}^*(du_Q) = (1 + {\iota _1}(f_Q))d({\iota _1}(u_Q)). \end{aligned}$$

  Since ${\iota _1}(u_Q)$ is again a local parameter at ${\iota _1}(Q)$ we have ${\iota _1}(u_Q) = cu_{{\iota _1}(Q)} + g_{{\iota _1}(Q)}$ where $c \ne 0$ and $v_{{\iota _1}(Q)}(g_{{\iota _1}(Q)}) > 1$. Therefore

  $$\begin{aligned} d({\iota _1}(u_Q)) = (c+ \frac{d g_{{\iota _1}(Q)}}{du_{{\iota _1}(Q)}})du_{{\iota _1}(Q)} \end{aligned}$$

  and

  $$\begin{aligned} -\Omega = (-1 - f_{{\iota _1}(Q)}) du_{{\iota _1}(Q)} = (1 + {\iota _1}(f_Q))(c +\frac{d g_{{\iota _1}(Q)}}{du_{{\iota _1}(Q)}})du_{{\iota _1}(Q)}. \end{aligned}$$

  Expanding the final product, one sees that $c = -1$ and $v_{{\iota _1}(Q)}(g_{{\iota _1}(Q)}) > n+1$.

• 2. This statement and proof are similar to [15, Lemma C.1]. Applying ${\iota _1}$ to (A.1), we have )

  $$\begin{aligned} -b = {\iota _1}(b)&= \frac{c_{Q,n}}{{\iota _1}(u_Q)^n} + \ldots + \frac{c_{Q,2}}{{\iota _1}(u_Q)^2} + \frac{c_{Q,1}}{{\iota _1}(u_Q)} + {\iota _1}(\tilde{f})\\&= \frac{(-1)^nc_{Q,n}}{u_{{\iota _1}(Q)}^n}(1+g_n) + \ldots + \frac{(-1)^2c_{Q,2}}{u_{{\iota _1}(Q)}^2}(1+g_2) \\&\quad + \frac{(-1)^1c_{Q,1}}{u_{{\iota _1}(Q)}}(1+g_1) + {\iota _1}(\tilde{f}) \end{aligned}$$

  where $v_{{\iota _1}(Q)}(g_\ell ) > n, n\ge \ell \ge 1$. This follows from the fact that ${\iota _1}(u_Q) = u_{{\iota _1}(Q)} +g_{{\iota _1}(Q)}, v_{{\iota _1}(Q)}(g_{{\iota _1}(Q)}) >n+1$ and so ${\iota _1}(u_Q)^{-\ell } = (-1)^\ell u_{{\iota _1}(Q)}^{-\ell } (1 + g_\ell )$ for some $g_\ell$ with $v_{{\iota _1}(Q)}(g_\ell ) > n$. Equating negative powers of $u_{{\iota _1}(Q)}$ yields the result.$\square$