DescriptionThe data below are given to facilitate checks in GP/PARI or Mathematica of the data given in the paper "Multiple polylogarithms in weight 4" by this author (arXiv:1609.05557).In particular we give the 122 term expression which we found for the 4-logarithm Li_4 whose `symbol' agrees, modulo products of lower weight terms, with the one for I_31(five term(x,y), z) . The existence of such an expression was conjectured in the early 90's by Goncharov. Furthermore we relate other functions of depth >1 to each other.
The 122 term expression(Mathematica-readable):
where tx[a] denotes the formal generator [a], while u[a,b,c,d] denotes the formal generator [cr(a,b,c,d)], where cr(a,b,c,d)=(a-c)(b-d)/( (a-d)(b-c)), involving the cross ratio of four arguments, and finally t[{j,..}] denotes the products of cross ratios cr_j(..) as defined in S.4.2 of the paper, as cr1[v_] := cr[v[[1]], v[[3]], v[[2]], v[[4]]]/ cr[v[[1]], v[[5]], v[[2]], v[[6]]]; cr2[v_] := cr[v[[1]], v[[2]], v[[3]], v[[4]]]/ cr[v[[1]], v[[2]], v[[5]], v[[6]]]; cr3[v_] := cr[v[[1]], v[[3]], v[[2]], v[[4]]]/ cr[v[[1]], v[[2]], v[[5]], v[[6]]]; cr4[v_] := cr[v[[1]], v[[3]], v[[2]], v[[4]]]/ cr[v[[1]], v[[2]], v[[5]], v[[6]]]* cr[v[[1]], v[[2]], v[[3]], v[[4]]]/ cr[v[[1]], v[[5]], v[[2]], v[[6]]]; cr5[v_] := cr[v[[1]], v[[2]], v[[3]], v[[4]]]/ cr[v[[1]], v[[3]], v[[2]], v[[4]]]* cr[v[[1]], v[[2]], v[[5]], v[[6]]]/ cr[v[[1]], v[[5]], v[[2]], v[[6]]]; cr6[v_] := cr[v[[1]], v[[3]], v[[2]], v[[4]]]/ cr[v[[1]], v[[6]], v[[2]], v[[5]]]* cr[v[[1]], v[[4]], v[[2]], v[[3]]]/ cr[v[[1]], v[[5]], v[[2]], v[[6]]];
A specialisation to e->ComplexInfinity, a->0, b->1 gives (or at least has the same symbol as)
Now take the five-fold sum (note that the second argument is inverted!) expr5term[x_, y_, z_, w_] := expr122spec[x, 1/y, z] + expr122spec[x, 1/y, w] + expr122spec[x, 1/y, (1 - z)/(1 - z*w)] + expr122spec[x, 1/y, 1 - z*w] + expr122spec[x, 1/y, (1 - w)/(1 - z*w)]; and check that the antisymmetrization expr5term[x, y, z, w] + expr5term[z, w, x, y] after antisymmetrising with respect to inversion of arguments, i.e. replacing L4[x_] by L[Factor[x]] - L[Factor[1/x]], has 931 terms with positive coefficient (we can ignore the ones with negative coefficient, due to the inversion relationfor the tetralogarithm given by L4[1/x] = -L4[x] modulo products) as follows:
The 931-term relationThe following combination gives the 931 terms in a functional equation for the tetralogarithm, in four variables x,y,z and w, as claimed in Corollary 19:
Now introducing the single-valued variant of the tetralogarithm (due to Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier) BW4[x_] := Im[Sum[2^j BernoulliB[j] /j! Log[Abs[x]]^(j) PolyLog[4 - j, x], {j, 0, 3}]] and plug it into experi[x_, y_, z_, w_] := expr5term[x, y, z, w] + expr5term[z, w, x, y] /. {L[xx_] :> BW4[xx],L4[xx_] :> BW4[xx]}; then a numerical check of the functional equation is that an arbitrary complex choice of the four variables should give numerically zero, e.g. experi[.1 + I, .2 - .3*I, .45 + .56*I, -.7 - .86*I] gives (the default precision in Mathematica rather small with about 14 digits) -2.13163*10^-14 Higher precisionFor more (say, 50) digits one can find on Mathematica stack exchange (highsciguy) the following suggestion: Theorem 20We also try to indicate how to numerically check Theorem 20 in the paper, this time in GP/PARI which for calculations is much better suited. In fact, we not only check it but rederive the coefficients given in that theorem. We first define the cross ratio as well as the nine functions f_i given in the theorem.
defines the cross ratio of four numbers, and the following defines the functions for the nine orbits in the text
then put all nine into a vector `ff'
We further invoke the S_5- and S_3-symmetries, as well as the sign of a permutation:
and define a single-valued cousin of Li_4 (due to Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier) which happens to be in-built in PARI
Further define the orbits under the action of S_5 x S_3 where S_5 acts on the first five arguments of any of the functions f1, f2, ..., f9 , and S_3 acts on the sixth argument (note the antisymmetry in both S_5 and S_3)--we imagine a vector A of length 5 of arbitrary complex numbers
Now produce the second argument "g" of "orbs" above using cross ratios from a second vector B of length 5 of arbitrary complex numbers -- due to the symmetries of the cross ratio we reduce to only 5 different values up to S_3-action, so invoking the full S_5 x S_5- symmetry on a pair (A, B) of vectors of length 5 we obtain
After all these preparations the crucial computation: we choose arbitrary entries for A and B, e.g.
and then compute the nine orbits, once with the order (A,B) , and a second time for (B,A), as we want to check that the corresponding sum over the nine orbits is indeed antisymmetric. This leaves us with two vectors of length nine, call them vl and vr for vector left and right, respectively (each one should take a couple of seconds).
For convenience we reproduce all the above nine `textareas' in one block which can simply be copy/pasted into GP/PARI:
The claim now is that, with appropriate integer coefficients, the sum of these gives zero. We can actually let PARI reproduce these vectors, using the very useful command `lindep(vec)' applied to a vector `vec' of real numbers which gives a linear combination that is `close to zero', and typically is used to numerically find integer relation among the numbers in `vec'. Here we ask for
and find (perhaps up to an overall sign)
which is indeed the vector of coefficients in Theorem 20. Now change the entries in A and B (say all different to avoid possible singularities) and try the same--you should always get these same nine coefficients.
ResourcesThe crucial package we used (currently private property of Duhr) produces symbols for the above objects and was implemented by C Duhr in the process of writing our joint paper (also with J Rhodes):From polygons and symbols to polylogarithmic functions, JHEP 10 (2012), 075.
|
email: Herbert Gangl