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**On poly(ana)logs. I. (With an appendix by Maxim Kontsevich: The \(1\frac{1}{2}\)-logarithm).**
*(English)*
Zbl 1062.11042

The authors investigate functional properties of the infinitesimal polylogarithms and the relationship between finite, infinitesimal and usual polylogarithms. The paper ends with a reproduction of a note of Kontsevich on the 1\(\frac{1}{2}\)-logarithm.

The \(n\)-th finite polylogarithm over a field \(F\) of characteristic \(p>0\) is the polynomial function \(\mathcal L_n(x) = \text{li}_n(x) = \sum_{k=1}^{p-1} x^k/k^n \in F[x]\). It was introduced for \(n=1\) by M. Kontsevich in an unpublished note (which is given as appendix of the present paper), in which he observed that the 1\(\frac{1}{2}\)-logarithm \(\mathcal L_1\) satisfies the same 4-term functional equation as the infinitesimal dilogarithm \(-(x\log|x| +(1-x)\log| 1-x|\). His explicit question for the next case was answered when it was found that the functional equation of J.-L. Cathelineau for the infinitesimal trilogarithm was also satisfied by \(\mathcal L_2\).

In the paper under review, the authors investigate the relationship between finite, infinitesimal and usual polylogarithms and prove directely several polynomial identities, such as the 22-term functional equation satisfied by \(\mathcal L_2(x)\), which is obtained from the Cathelineau identity for the infinitesimal trilogarithm. As a matter of fact, the general machinery used to deduce the functional equation for higher \(\mathcal L_{n-1}\) from functional equations for infinitesimal \(n\)-logarithms relies on the following three main features:

(i) The fact that the \(p\)-adic polylogarithms introduced by R. Coleman [Invent. Math. 69, 171–208 (1982; Zbl 0516.12017)] satisfy the same functional equations as the classical ones, as proved by Z. Wojtkowiak [Bull. Soc. Math. Fr. 119, 343–370 (1991; Zbl 0748.12006)].

(ii) The tangential procedure of J.-L. Cathelineau [Ann. Inst. Fourier 46, 1327–1347 (1996; Zbl 0861.19003)], which yields an analogue for infinitesimal polylogarithms.

(iii) The results of A. Besser [Compos. Math 130, 215–223 (2002; Zbl 1062.11041)], who proved that, for \(p>n+1\), the finite \(n\)-logarithm is the reduction \(\mod p\) of an infinitesimal \(p\)-adic \(n+1\)-logarithm.

The organization of the paper is as follows: Part I is devoted to preliminary background on classical and infinitesimal polylogarithms. Part II introduces the finite polylogarithms, gives their functional equations and investigates their characterizations. The main proofs are given in Part III. Finally the Appendix reproduces the note of Kontsevich.

The \(n\)-th finite polylogarithm over a field \(F\) of characteristic \(p>0\) is the polynomial function \(\mathcal L_n(x) = \text{li}_n(x) = \sum_{k=1}^{p-1} x^k/k^n \in F[x]\). It was introduced for \(n=1\) by M. Kontsevich in an unpublished note (which is given as appendix of the present paper), in which he observed that the 1\(\frac{1}{2}\)-logarithm \(\mathcal L_1\) satisfies the same 4-term functional equation as the infinitesimal dilogarithm \(-(x\log|x| +(1-x)\log| 1-x|\). His explicit question for the next case was answered when it was found that the functional equation of J.-L. Cathelineau for the infinitesimal trilogarithm was also satisfied by \(\mathcal L_2\).

In the paper under review, the authors investigate the relationship between finite, infinitesimal and usual polylogarithms and prove directely several polynomial identities, such as the 22-term functional equation satisfied by \(\mathcal L_2(x)\), which is obtained from the Cathelineau identity for the infinitesimal trilogarithm. As a matter of fact, the general machinery used to deduce the functional equation for higher \(\mathcal L_{n-1}\) from functional equations for infinitesimal \(n\)-logarithms relies on the following three main features:

(i) The fact that the \(p\)-adic polylogarithms introduced by R. Coleman [Invent. Math. 69, 171–208 (1982; Zbl 0516.12017)] satisfy the same functional equations as the classical ones, as proved by Z. Wojtkowiak [Bull. Soc. Math. Fr. 119, 343–370 (1991; Zbl 0748.12006)].

(ii) The tangential procedure of J.-L. Cathelineau [Ann. Inst. Fourier 46, 1327–1347 (1996; Zbl 0861.19003)], which yields an analogue for infinitesimal polylogarithms.

(iii) The results of A. Besser [Compos. Math 130, 215–223 (2002; Zbl 1062.11041)], who proved that, for \(p>n+1\), the finite \(n\)-logarithm is the reduction \(\mod p\) of an infinitesimal \(p\)-adic \(n+1\)-logarithm.

The organization of the paper is as follows: Part I is devoted to preliminary background on classical and infinitesimal polylogarithms. Part II introduces the finite polylogarithms, gives their functional equations and investigates their characterizations. The main proofs are given in Part III. Finally the Appendix reproduces the note of Kontsevich.

Reviewer: Jean-FranĂ§ois Jaulent (Talence)

### MSC:

11G55 | Polylogarithms and relations with \(K\)-theory |

19F15 | Symbols and arithmetic (\(K\)-theoretic aspects) |

33B30 | Higher logarithm functions |