Tougher groups (4/4)
Not all signatures are afforded by this construction. The signature is constant between
consecutive real roots of poldisc(P), and may change when passing a root. Continuing the
previous example in degree 12:
? D = poldisc(Pt);
? D /= gcd(D, D’) \\ make it squarefree
? polrootsreal(D)
%3 = [-367.2, -71.5, -1.4, -7.E-6, -6.8E-6, 0, 6.4E-4, 1.0, 50.8]
? polsturm(subst(D,’t,-1)) \\ returns r1
%4 = 0
? polsturm(subst(D,’t, 2))
%5 = 4
Checking rational values of t between the above roots, we see that the signature is (0, 6) for
t < 1 and (4, 4) for t > 1.
PARI/GP day 2021 (02/06/2021) – p. 14/14