2. THE TODA HIERARCHY 13
and, since z C is arbitrary, obtains the Burchnall-Chaundy polynomial (see [16],
[17] in the case of differential expressions) relating P2P+2 and L,
23+1
(2.48) P22,+2 = R29+2(L) = J J (L - Em).
m=0
The resulting hyperelliptic curve Kg of (arithmetic) genus g obtained upon com-
pactification of the curve
2?+l
(2.49) y2 = R2g+2{z) =]\(z- Em)
m=0
will be the basic ingredient in our algebro-geometric treatment of the Toda and
Kac-van Moerbeke hierarchies in the remainder of this exposition.
The spectral theoretic content of the polynomials Fg and Gg+\ is clearly dis-
played in (4.8), (4.19),(4.20) and especially in (4.32)-(4.36).
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