Elementary symmetric functions of reciprocals of monic polynomials in function fields

Question

Let $q$ be a prime power and $\mathbb{F}_q$ the field of cardinality $q$. Let $A = \mathbb{F}_q[T]$ and let $A_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A_+$ and let $k$ be a positive integer. Set $$e(k) = \sum_{\begin{matrix} a_1, a_2, \ldots, a_k \in A_+ \\ a_1<a_2<\cdots<a_k \end{matrix}} \frac{1}{a_1 a_2 \cdots a_k}$$ where the sum is considered in the $T^{-1}$-adic topology. In non-archimedean analysis there is no need to discuss the order of summation.

I'm trying to find out what is known about algebraic relations between the $e(k)$ over $\mathbb{F}_q(T)$. Here is what I have found so far:

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The $e(k)$ are very close to the Taylor coefficients of Thakur's geometric $\Gamma$ function: $$\tfrac{1}{\Gamma(z)} = z \prod_{a \in A_+} (1+z/a).$$ (See Thakur, 1991, Section 5.) But the papers I found in a quick search concentrate on evaluating $\Gamma$ at points of $\mathbb{F}_q(T)$, not the coefficients of the Taylor series.

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The sum $\zeta(k) = \sum_{a \in A_+} \tfrac{1}{a^k}$ is in the ring generated by the $e(k)$. This sum is called the Goss $\zeta$-function and is very well understood: We have the relations $\zeta(kp) = \zeta(k)^p$ and $\zeta((q-1) m) = B(m) \pi^{(q-1)m}$ (for positive integer $m$) where $B(m)$ is an element of $\mathbb{F}_q(t)$ analogous to a Bernoulli number and $\pi$ is a transcendental element of $\mathbb{F}_q[[(-T)^{-1/(q-1)}]]$. Other than these relations, the $\zeta(m)$ are algebraically independent over $\mathbb{F}_q(T)$. See Yu, 1991.

Newton's identities show that the ring generated by the $e(k)$ is the same as the ring generated by the $\zeta(k)$ and the $e(pk)$. Obviously, this leaves room for the ring generated by the $e(k)$ to be much larger.

Is the ring generated by the $e(k)$ actually this much larger, or am I missing a trick?

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Chang 2012 and others study $$\zeta(s_1, s_2, \ldots, s_k):= \sum_{\deg(a_1) < \deg(a_2) < \cdots < \deg(a_k)} \frac{1}{a_1^{s_1} \cdots a_k^{s_k}}.$$

Is this version of multiple $\zeta$ values related to my question, or is it unrelated?

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If we remove the monic condition in the summation, then the sum is of the form $\pi^k Q(k)$ for some $Q(k) \in \mathbb{F}_q(T)$. This is Proposition 2.1 in this note of mine.

The reason I started going into this question is that I am spending today adding references to this paper in preparation to putting it on the arXiv and sending it off.

Answer 1

Many of the various types of function field valued multiple zeta values (MZV's) were first defined by Dinesh Thakur in his book "Function Field Arithmetic" from 2004 (see section 5.10). He considers several possible definitions, for example $$ \zeta_l(s_1, \ldots, s_k) = \sum_{\substack{a_1, \ldots, a_k \in A_+ \\ a_1 > \cdots > a_k}} \frac{1}{a_1^{s_1} \cdots a_k^{s_k}} \in \mathbb{F}_q((T^{-1})), $$ where the ordering is an arbitrarily chosen lexicographic ordering of $A_+$ and $s_1, \ldots, s_k$ are positive integers. It seems then that your value $e(k)$ is $\zeta_l(1, \ldots, 1)$ of depth $k$. He also defines $$ \zeta_d(s_1, \ldots, s_k) = \sum_{\substack{a_1, \ldots, a_k \in A_+ \\ \deg a_1 > \cdots > \deg a_k}} \frac{1}{a_1^{s_1} \cdots a_k^{s_k}}, $$ which is the type of MZV you mentioned studied by Chang. He also defines a couple of other variants.

To address one of your questions, from what Thakur says in Remark 5.10.16, it would appear that the $e(k)$'s are probably not algebraically related to the MZV's $\zeta_d(s_1, \dots, s_k)$.

Anderson and Thakur ("Multizeta values for $\mathbb{F}_q[t]$, their period interpretation, and relations between them", IMRN, 2009) showed that the MZV's $\zeta_d(s_1,\dots, s_k)$ defined using degrees can be realized as periods of certain higher dimensional Drinfeld modules (Anderson $t$-modules) that are iterated extensions of tensor powers of the Carlitz module. They then can also be expressed as linear combinations of values of Carlitz multiple polylogarithm functions. Using this period interpretation various bits of transcendence machinery can be applied, which is the subject of the paper you mention by Chang (Compositio, 2014).

Thakur mentions in that same remark 5.10.16 that the lexicographic MZV's $\zeta_l(s_1, \dots, s_k)$ do not have natural relations with periods of Drinfeld modules, so at least on the surface they seem to represent a different class of numbers than the $\zeta_d(s_1, \dots, s_k)$'s. Thakur follows this up with numerical calculations when $q=2$ and shows that they do not appear to be rational multiples of Carlitz zeta values.

Of course this does not answer your question definitively, but I think it would be safe to say that it is not known whether there are any general relations.

Also, one minor correction: when you remark that it was shown that the Carlitz zeta values $\zeta(m)$ are algebraically independent over $\mathbb{F}_q(T)$ (aside from the known relations involving Bernoulli-Carlitz numbers and $p$-th powers), this result is actually due to Chang and Yu ("Determination of algebraic relations among special zeta values in positive characteristic," Adv. Math. 216 (2007), 321-345). In the 1991 paper of Yu, he proves that the values are all transcendental over $\mathbb{F}_q(T)$, but the algebraic independence questions were answered later.