README for directory "S-invariant calculations"

This directory contains a number of Mathematica notebooks, 
created by Mathematica 4.2.1, which perform the calculations
described in the manuscript "Some geometry and combinatorics 
for the S-invariant of ternary cubics".  They were run on 
a Sun V880 at the Max Planck Institute for Mathematics with 
8 CPUs and 16GB of Main Memory theoretically available 
(of which probably only a fraction was used by these programs).

Most of the notebooks refer to the check (mentioned in Section 2 
of the paper) that the formal $S$-invariant considered there as 
a polynomial in the variables a_1, ..... , a_r, b_1 , .... , b_r,
c_1, .... , c_r, where r = 3d-3, trihomogeneous of degree 4d-4 
in each set of variables, only has POSITIVE coefficients for the 
cases d \le 5.  This is checked for each monomial, each of which 
corresponds to a 3 by (4d-4) matrix; the jth column denoting 
the powers of a_j, b_j, c_j which occur (the sum being 4).  Thus, 
if the first column is (2,1,1), this denotes that a_1 ^2 b_1 c_1 
occurs in the monomial.

The notebook S444.nb deals with the case when d \le 4; this is 
the easy case, and so only one sample calculation is included.  
This may also be regarded as the case d=5 with at least one a_i,
at least one b_j and at least one c_k occurring in the given 
monomial with power 4.  The coefficients are obtained by successive
differentiation, and since we are aiming only to check positivity of
the coefficients, considerations of symmetry cut down considerably
the number of cases needing to be checked.  The differentiations 
are performed in nested series of loops.

For the remaining cases with d=5, we first calculate and store the
function $S$ as a function of the a_i,b_j and c_k.  We do not 
expand this function, as the resulting polynomial would be totally 
unmanageable.  This is done in the notebook S555.nb, which uses
the formula for S as a sum of 25 terms.  Slightly better, in 
S555Alt.nb, we use the formula for S given in Section 1 of the 
paper in terms of minors of the Jacobian matrix --- in the 
subsequent calculations, this turns out to be marginably more 
efficient.  In both notebooks, the function S is saved in a 
file called tsave.

We now need to check the positivity of coefficients for all the
various shapes of monomials.  This is done in the notebooks with
names of the form Checkxxx.nb.  In each case, the the formula 
obtained for S should be read in from tsave and denoted as T, and then 
all necessary coefficients are calculated by successive differentiation
and checked to be positive.  Thus Check44.nb deals with the case 
when one row of the 3 \times 12 matrix of exponents contains two 
or more entries 4 (and at most one of the other rows contains a 4).
The other notebooks are labelled similarly, and the cases being 
dealt with are summarized at the start of each notebook.  The final
case is Check2.nb, which is the relatively simple case where only 
2, 1 or 0 appear in the matrix of exponents.

The remaining notebooks consist of various formulae referred to  
in the paper.  For instance Formula.nb is the formula appearing
in the Appendix to the paper --- because this formula is of 
particular interest, a tex file of it is also included, labelled 
Formula.tex.  Formula3.nb gives the derivation of the formula for the 
coefficient of the simple monomial which appears in the penultimate
example of the paper.  The first derivation is via a simple minded 
algorithm, and the second derivation via the better algorithm 
described in Section 4.  In both cases, the formula simplifies to 
the very simple form described in Section 4.