The Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands having only simple affine poles [1, 2, 7]. While it is very effective in the case of simple radical extensions, its use in more general algebraic extensions requires the precomputation of an integral basis, which makes the reduction impractical for either multiple algebraic extensions or complicated ground fields. In this work, Manuel Bronstein shows that the Hermite reduction can be performed without a priori computation of either a primitive element or integral basis, computing the smallest order necessary for a particular integrand along the way.
w= 

(a_{1}w_{1}+...+a_{n}w_{n}) 
M= 

N= 

m_{i}= 


b_{i,j}w_{j} for 1£ i£ n. 
f= 

æ ç ç è 

M_{w}^{t}mG_{w}UV'I_{n} 
ö ÷ ÷ ø 

=G_{w}S^{1} 

(2) 
w= 


T_{i}w_{i}Î O. 

A_{i}w_{i}= 


T_{i}S_{i}. 
w= 


T_{i}w_{i}Î O. 

A_{i}w_{i}= 

h_{i}S_{i} 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.