Symbolic and numerical computation in determining P-critical unit forms and Tits P-critical posets

We define algorithms constructing P-critical unit quadratic forms q : Z n → Z from positive ones p : Z n − 1 → Z with a sincere root, up to an action of the orthogonal group O ( n , Z ) , where Z is a ring of integers. We apply them to compute P-critical forms and positive forms, for n ⩽ 9 , as well as one-peak posets I with P-critical Tits form q ˆ I : Z I → Z . By applying symbolic computations in Maple and numerical in C# we show that every such one-peak poset has at most nine elements, we compute a complete list L 1 , … , L 116 of one-peak posets I with P-critical Tits form, and we determine their Coxeter types Δ I that are simply-laced Euclidean diagrams A ˜ n , D ˜ n , E ˜ 6 , E ˜ 7 , E ˜ 8 .