P-critical integral quadratic forms and positive unit forms: An algorithmic approach

We introduce the class of P-critical integral unit forms q:Zn→Z containing the critical forms in the sense of Ovsienko [13]. Several characterisations of P-critical forms are given. In particular, it is proved that q is P-critical if and only if there is a uniquely determined extended Dynkin diagram and a special group isomorphism T:Zn→Zn such that q∘T is the quadratic form qΔ:Zn→Z,n=|Δ0|, of the diagram Δ. A correspondence between positive forms p:Zn-1→Z with a sincere root and P-critical forms q:Zn→Z is described and efficient linear algebra algorithms for computing P-critical unit forms and positive forms are constructed.