From jeroos@matematik.su.se Tue Nov  2 19:48:20 1999
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From: Jan-Erik Roos <jeroos@matematik.su.se>
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Date: Tue, 2 Nov 1999 18:36:28 +0100
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To: jeroos@pavidus.matematik.su.se, joeb@pavidus.matematik.su.se,
        ufn@pavidus.matematik.su.se, ufn@maths.lth.se
Subject: test_cases
Status: RO

Hi, here are some test cases for bergman and for anick:

Here is a case that gives different results in all characteristic:

A = k<x,y,z,u,v,w>/
([x,v],[x,w],[y,u],[y,v],[y,w],[z,v],[x,u]-[z,w],[x+w,z+u]>
Here [x,v] =x*v+v*x, etc.
The input file in LISP-form should be (if the characteristic is 3):

(setq embdim 6)
(setmodulus 3)
(LISPFORMINPUT)
((1 1 5)(1 5 1))
((1 1 6)(1 6 1))
((1 2 4)(1 4 2))
((1 2 5)(1 5 2))
((1 2 6)(1 6 2))
((1 3 5)(1 5 3))
((1 1 4)(1 4 1)(-1 3 6)(-1 6 3))
((1 1 3)(1 3 1)(1 1 4)(1 4 1)(1 6 3)(1 3 6)(1 6 4)(1 4 6))
(LISPFORMINPUTEND)
% The Hilbert series A(z) should, if the characteristic is p, be
% given by the formula
% 1/A(z) = (1-6*z+8*z^2)-z^(p+2)/(1-z^p)
% Furthermore, if you apply ANICK on this algebra you should get
% that the global dimension is 3, but the Tor_3:s should be such
% that the Hilbert series is as above (i.e. the Tor_3:s are of
% dimension 1 in different degrees , which are  p+2,2p+2,3p+2,...).


% Finally here is a test case where I am unable to determine what the
% Hilbert series should be (I could as you se calculate rather high up):

(setq embdim 5)
(LISPFORMINPUT)
((1 1 1))
((1 1 2)(1 2 1))
((1 1 3)(1 3 1))
((1 2 3)(1 3 2))
((1 3 3))
((1 3 4)(1 4 3))
((1 1 4)(1 4 1)(-1 2 5)(-1 5 2))
((1 2 4)(1 4 2)(-1 3 5)(-1 5 3))
((1 4 5)(1 5 4)(-1 5 5))
(LISPFORMINPUTEND)
% Here the result for the first coefficients of the Hilbert series are:
% 1,5,16,44,110,257,572,1230,2581,5320,10824,21820,43704 but what are
% the higher terms up to degree 20 ? 
% Do we have an expansion of something irrational here ?

Best regards
Jan-Erik

