There is the following algorithm:

First of all, at the first line do they mean that the content of j is i? About the second line, why don't we subtract the polynomial f[i]*uhX^i ? Also, what do they mean at the part In line 5.... unaltered. How do we see this? @SEJPM

@Evinda yes, this is a for-loop and the initial value of the inner counter variable j is i

@Evinda as for the line 5 thingy, I think they are stating the loop invariant, ie the logical expression that stays true for all executions

@Evinda as for the second line thingy, I think that the i-d comes from the line 7 where you don't go below i-d and this makes sense (kinda) when you remember that d is the degree of the polynomial and i is the current monomial you are trying to remove (?)

@SEJPM Is there then maybe an error at the algorithm at the line 8? Because at the first iteration for i=d' and j=i=d' we would get f[d']<-f[d']-uf[d']h[d'], but h[d'] is 0, when we suppose that d<d'. So should it maybe be f[j]<-f[j]-ah[j-d] ?

@SEJPM As for this, at line 5, we have a<-u*f[i]. Why is it said that the polynomial $f[i] \cdot u \cdot h \cdot X^{i-d}$ is added to $h \cdot (q[0], \dots, q[d'-d])$ ?

@Evinda I think this is what happens in line 6

Why is the polynomial added to $h \cdot (q[0], \dots, q[d'-d])$ at line 6? :/ @SEJPM