I have some trouble following an example for the f4-algorithm in the attached paper. The example is given under point 2.6 on page 14 and 15. The example is supposed to show how the improved F4 algorithm (page 9) works. I am concerned about the content of the set P, which only seems to change in two cases: 1) the update algorithm is invoked (I have attached that too), 2) an element from P is chosen by the Sel()-function. In particular the example states that an element (f_2,f_4) is chosen during the second run through the second while loop of the f4 algorithm (unfortunately the indices do not agree with the indices the polynomials are assigned during the definition of set F at the beginning of the example, f_1 is f_4, f_2 is f_3, f_3 is f_4 and f_4 is f_1 in the original definition). I do not have any such pair in P when doing the calculation by pen and paper. When entering the second while loop for the first time P contains the following pairs: (f_4,f_3), (f_3,f_2),(f_2,f_1) (indices according to what is used in the example's main text, i.e. f_1=abcd-1). The pair (f_4,f_3) is chosen by Sel(P) and no new pair is added during the next call to the update algorithm. So in my book P only contains the pairs (f_3,f_2),(f_2,f_1) when the second while loop is entered for a second time. Does anyone have an idea where the pair (f_2,f_4) in the example comes from?