Umemura's formula expresses the roots of an algebraic equation in terms of theta constants whose period matrix is obtained from Abelian integrals on a hyperelliptic curve (Umemura 1984).
Let 
be an irreducible polynomial of polynomial
degree 
over a subfield of 
. Form the hyperelliptic
curve 
, where
 |
(1)
|
Its genus is 
, where 
denotes the floor function.
Let 
, ..., 
be the roots of
. Order the finite branch
points as 0, 1, 
, ..., 
, followed by 2 when 
is even, and choose a compatible basis of homology
for the normalized period matrix 
. Put 
and 
, and write 
for the
fourth power of a theta constant. Put 
, 
, and 
. Then
 |
(2)
|
The characteristics in this expression must be used with the compatible ordering and period basis. Changing the ordering gives the other roots (Umemura 1984; Shokurov 1998, p. 231). The expression uses Abelian integrals and theta constants, so it is not a solution by radicals.
