The data package contains the following data:
- S168 ... the BMV polynomial S_{16,8}(X^2,Y^2)
- V ... the vector of all words of order 16, which can appear in an
SOHS polynomial cyclically equivalent to S168.
Note that V=[V1;V2;V3], where Vi is a vector as in Theorem 2.1
- G0 = a (block diagonal) Gram matrix for S168 - the one we used in (10).
- A ... a matrix of order 19600 x 440 ... each column of A corresponds
to equations as in (7) or (8)
- C ... a matrix of order 19600 x 3305 ... columns of C are pairwise
orthogonal and also orthogonal to columns of A ... matrix reformulations
of columns of C are exactly matrices Ci from (10).
Note: we kept in A and C only columns which corresponds to the diagonal
blocks, as described in Lemma 2.3, hence we have in A and C altogether
70*71/2 + 2*35*36/2 = 3745 columns.
- B ... a solution of (10). Note that B = blockDiag(B1,B2,B3) - a PsD matrix
of order 140x140.
Instructions (some of this requires NCSOStools):
1. To reproduce the polynomial S_{16,8}(X^2,Y^2), run
S168=BMVq(16,8);
2. To check that G0 is a Gram matrix for S168 call
Snew=V'*G0*V; NCisCycEq(S168,Snew)
(Caution: the last command must give answer 1 and takes quite some time
to evaluate.)
3. To check that A contains the equations (7) compute
trace(reshape(A(:,i),140,140)*G0)
which must be the number of all words in S168, which are cyclically
equivalent to the words w or w^*, underlying the i-th equation
4. To check that B is feasible for the linear constraints in (10) run
norm(C'*B(:))==0, trace(B*G0)<0
Note that trace(B*G0)=-8 and not -100 as stated in (10) - the reason is
that B is integer matrix obtained by rounding and projecting and rescaling.
5. B is an integer matrix. To see that is it PsD, compute
min(eig(B))>0
Alternatively, for a symbolic verification, please use our Mathematica
notebook
bmv_16_8-ldlt.nb
available from http://ncsostools.fis.unm.si where the LDU factorization
is given.