Notes on the recorded outputs in this folder (added when the note was written; the .out files are unchanged).

conj_check_parts12.out   Parts 1 and 2 of conj_check.py (the full run was stopped during an earlier, larger
                         Part 3; Part 3 was then run alone with the present parameters).
conj_check_part3.out     Output of "python3 -I conj_check.py part3". Only its line "Part 3, 20000 random
                         products ..." is a result. The line "Part 2, all ULC sequences with h_0=1, 1<=h_i<=0,
                         2<=d<=0: 0 sequences ..." and the last line "Total real-rooted polynomials tested
                         exhaustively in Part 1: 0" are artefacts of the option part3, which skips Parts 1 and 2.
                         A full run without argument prints the same Part 3 line and the total 11089.
conj_check_aborted_run1.out  First run with a larger Part 2 (h_i <= 80, d <= 12), stopped after Part 1 because it
                         was too slow; its Part 1 lines agree with conj_check_parts12.out.
lemma_check.out          Its "min ratio" is ((k+1)! a^<k>)^k / (k! a)^(k+1), the k-th power of a^<k>/Phi_k(a);
                         this is why its minimum is reached at k = 1 (see Remark 6.1 of the note).

Degree coverage of Part 1: the real-rooted polynomials found in the boxes of degree 5, 6 and 7 (h_i <= 12, 7, 5)
are only (1+t)^5; the Macaulay conditions with k >= 2 are exercised mainly by Part 2 (all (ULC) sequences with
h_i <= 30, d <= 10) and Part 3 (random products of degree up to 25).
