Algebraic independence results for colored multizeta values in characteristic \(p\). (English) Zbl 1525.11093
The paper under review deals with colored multizeta values (CMZs) and multipolylogarithms (CMPs) in positive characteristic. More precisely, these multizeta values are elements of \(\mathbb{F}_q (( \frac{1}{\theta} ))\), and generalize D. S. Thakur’s definition in the book entitled Function field arithmetic [River Edge, NJ: World Scientific (2004; Zbl 1061.11001)]. The author first gives a reinterpretation of CMPs as periods of mixed Carlitz motives (Proposition 4.3), and he rewrites CMZs as \(k\)-linear combinations of CMPs. Then, he studies linear independence of CMPs by using the techniques developed in [M. A. Papanikolas, Invent. Math. 171, No. 1, 123–174 (2008; Zbl 1235.11074)], and applies this to infer algebraic independence of large classes of CMZs, defined by
\[
\zeta_{s_1\ldots, s_r} (\varepsilon_1,\ldots, \varepsilon_r)= \sum_{\stackrel{\deg a_1 >\ldots \deg a_r \geq 0}{a_i\in A_+} } \frac{\varepsilon_1^{\deg a_1} \cdots \varepsilon_r^{\deg a_r}}{a_1^{s_1}\cdots a_r^{s_r}},
\]
where \(s_i\) are positive integers, \(\varepsilon_i \in \mathbb{F}_q^\times\) and \(A_+\) denotes the set of monic polynomials in \(\mathbb{F}_q[\theta ]\).
Reviewer: Sami Omar (Sukhair)
MSC:
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
| 11G09 | Drinfel’d modules; higher-dimensional motives, etc. |
References:
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