88
N.M. Katz:
Then the matrix of f(a, b, c) on N is
/P(O)
Q(O)
P(1)
Q(I!.
(6.4.0.9)
t @(2)
." ." ." .
.."
'.
It's proper values are thus its diagonal terms. If a = b, then all but one of
these diagonal terms is nonzero, and hence zero is a proper value of
multiplicity one, which implies that the kernel of g( a,  b,  c) in N is
one dimensional. Thus we may assume a +b, and that a< b, since the
problem is symmetric in a and b.
As a + b, the matrix (6.4.0.9) has precisely two diagonal terms which
vanish, namely
P(a)
and
P(v).
The subspace
N(<b)
of N consisting of
polynomials of degree <b is stable under (, and f is invertible on
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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